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<p><strong>Description:</strong> Covers strong induction as a tool for proofs. Introduction to invariants with different games, including the n–block game and grid puzzles.</p> <p><strong>Speaker:</strong> Tom Leighton</p>
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<h2 class="subhead">Related Resources</h2>
<p>Top Ten Proof Techniques Not Allowed in 6.042 (<a href="../../../contents/video-lectures/lecture-3-strong-induction/MIT6_042JF10_proof.pdf" target="_blank">PDF</a>)</p>
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<p><span m="570">The</span> <span m="680">following</span> <span m="1120">content</span> <span m="1710">is</span> <span m="1830">provided</span> <span m="2270">under</span> <span m="2550">a</span> <span m="2590">Creative</span> <span m="2990">Commons</span> <span m="3400">license.</span> <span m="4510">Your</span> <span m="4700">support</span> <span m="5200">will</span> <span m="5360">help</span> <span m="5600">MIT</span> <span m="6060">OpenCourseWare</span> <span m="6850">continue</span> <span m="7360">to</span> <span m="7440">offer</span> <span m="7850">high-quality</span> <span m="8610">educational</span> <span m="9240">resources</span> <span m="9860">for</span> <span m="10010">free.</span> <span m="11220">To</span> <span m="11320">make</span> <span m="11430">a</span> <span m="11470">donation</span> <span m="12160">or</span> <span m="12430">view</span> <span m="12870">additional</span> <span m="13290">materials</span> <span m="13830">from</span> <span m="13980">hundreds</span> <span m="14410">of</span> <span m="14520">MIT</span> <span m="14950">courses,</span> <span m="16070">visit</span> <span m="16280">MIT</span> <span m="16700">OpenCourseWare</span> <span m="17740">at</span> <span m="17910">ocw.mit.edu.</span> </p>
<p><span m="23250">PROFESSOR: Now</span> <span m="23400">generally</span> <span m="23830">speaking,</span> <span m="24290">a</span> <span m="24360">proof</span> <span m="24700">is</span> <span m="24820">going</span> <span m="24930">to</span> <span m="25000">have</span> <span m="25900">seven</span> <span m="26330">characteristics</span> <span m="28226">that</span> <span m="28670">you</span> <span m="28850">want</span> <span m="29010">to</span> <span m="29080">keep</span> <span m="29290">in</span> <span m="29390">mind.</span> <span m="33950">Good</span> <span m="34180">proofs</span> <span m="35200">are</span> <span m="36860">correct--</span> <span m="38350">that's</span> <span m="38810">obviously</span> <span m="39370">important--</span> <span m="41470">complete--</span> <span m="45140">you've</span> <span m="45240">got</span> <span m="45410">to</span> <span m="45470">have</span> <span m="45670">all</span> <span m="45930">the</span> <span m="46020">details</span> <span m="46630">there.</span> <span m="46930">All</span> <span m="47260">of</span> <span m="47460">the</span> <span m="47550">key</span> <span m="47780">steps</span> <span m="48160">have</span> <span m="48360">to</span> <span m="48450">be</span> <span m="48560">there.</span> </p>
<p><span m="50680">They</span> <span m="50830">should</span> <span m="51000">be</span> <span m="51130">clear</span> <span m="52620">so</span> <span m="52780">we</span> <span m="52880">can</span> <span m="52990">understand</span> <span m="53630">what's</span> <span m="53880">going</span> <span m="54140">on.</span> <span m="58070">Brief</span> <span m="58440">is</span> <span m="58600">good.</span> <span m="60300">You</span> <span m="60590">don't</span> <span m="60710">want</span> <span m="60850">to</span> <span m="60920">crush</span> <span m="61310">somebody</span> <span m="61890">with</span> <span m="62260">all</span> <span m="62530">the</span> <span m="63120">details.</span> <span m="63630">You</span> <span m="63730">want</span> <span m="63890">to</span> <span m="64260">get</span> <span m="64450">to</span> <span m="64530">the</span> <span m="64629">key</span> <span m="64830">points,</span> <span m="65459">and</span> <span m="65630">be</span> <span m="65750">crisp.</span> </p>
<p><span m="68570">It's</span> <span m="68770">really</span> <span m="69080">nice</span> <span m="69400">if</span> <span m="69510">they're</span> <span m="69980">elegant.</span> <span m="73730">Now,</span> <span m="74910">that</span> <span m="75100">means</span> <span m="75330">clever.</span> <span m="76320">It's</span> <span m="76630">the</span> <span m="76730">mathematician's</span> <span m="77750">notion</span> <span m="78200">of</span> <span m="78300">beauty.</span> </p>
<p><span m="79470">Like</span> <span m="79600">you</span> <span m="79680">go</span> <span m="79850">to</span> <span m="79920">the</span> <span m="80070">art</span> <span m="80250">museum,</span> <span m="80880">and</span> <span m="80950">the</span> <span m="81030">artists</span> <span m="81290">will</span> <span m="81390">say,</span> <span m="81730">wow,</span> <span m="82100">that's</span> <span m="82330">a</span> <span m="82400">beautiful</span> <span m="82830">painting.</span> <span m="84210">In</span> <span m="84350">mathematics,</span> <span m="84990">you</span> <span m="85080">say,</span> <span m="85650">wow,</span> <span m="85930">that's</span> <span m="86150">an</span> <span m="86270">elegant</span> <span m="86820">proof.</span> <span m="88160">It's</span> <span m="88570">crisp,</span> <span m="89460">clever,</span> <span m="90040">short,</span> <span m="90530">to</span> <span m="90610">the</span> <span m="90710">point.</span> </p>
<p><span m="91410">And</span> <span m="91590">it's</span> <span m="91730">really</span> <span m="92020">the</span> <span m="92310">highest</span> <span m="92600">compliment</span> <span m="93040">you</span> <span m="93160">can</span> <span m="93310">get</span> <span m="94780">from</span> <span m="94960">a</span> <span m="95000">mathematician,</span> <span m="95570">anyway.</span> <span m="96770">And</span> <span m="96850">there's</span> <span m="96980">a</span> <span m="97030">lot</span> <span m="97220">of</span> <span m="97280">judgment</span> <span m="97770">that</span> <span m="97840">goes</span> <span m="98070">into</span> <span m="98250">that.</span> <span m="101090">Just</span> <span m="101310">like</span> <span m="101470">in</span> <span m="101580">art--</span> <span m="101810">there's</span> <span m="101980">judgment</span> <span m="102340">over</span> <span m="102490">what</span> <span m="102690">is</span> <span m="103070">great</span> <span m="103330">art.</span> </p>
<p><span m="104570">The</span> <span m="104690">proof</span> <span m="104920">should</span> <span m="105080">be</span> <span m="105170">well</span> <span m="105480">organized.</span> <span m="110690">For</span> <span m="110810">example,</span> <span m="111280">use</span> <span m="111570">lemmas</span> <span m="112280">the</span> <span m="112390">same</span> <span m="112740">way</span> <span m="112950">you</span> <span m="113060">would</span> <span m="113170">use</span> <span m="113380">subroutines</span> <span m="114170">in</span> <span m="114250">writing</span> <span m="114500">code--</span> <span m="116130">helps</span> <span m="116400">to</span> <span m="116470">make</span> <span m="116620">it</span> <span m="116710">clear.</span> <span m="118400">And</span> <span m="118690">the</span> <span m="118790">proof</span> <span m="119080">should</span> <span m="119240">be</span> <span m="119440">in</span> <span m="119760">order.</span> </p>
<p><span m="123150">Sometimes</span> <span m="123630">you'll</span> <span m="123750">see</span> <span m="123960">proofs</span> <span m="124350">where</span> <span m="124480">things</span> <span m="124960">are</span> <span m="125030">done</span> <span m="125240">in</span> <span m="125340">a</span> <span m="125370">haphazard</span> <span m="126440">fashion.</span> <span m="126950">All</span> <span m="127160">the</span> <span m="127260">pieces</span> <span m="127660">are</span> <span m="127710">there,</span> <span m="127920">but</span> <span m="128090">they're</span> <span m="128289">in</span> <span m="128350">the</span> <span m="128460">wrong</span> <span m="128759">order.</span> <span m="129810">Sometimes--</span> <span m="130660">and</span> <span m="130729">they</span> <span m="130850">teach</span> <span m="131190">this</span> <span m="131390">in</span> <span m="131490">some</span> <span m="131740">high</span> <span m="131960">schools--</span> <span m="133120">they</span> <span m="133260">teach</span> <span m="133590">you</span> <span m="133670">to</span> <span m="133730">do</span> <span m="133950">proofs</span> <span m="134870">backwards.</span> </p>
<p><span m="136790">And</span> <span m="136870">the</span> <span m="136940">classic</span> <span m="137430">thing</span> <span m="137640">there</span> <span m="138000">is,</span> <span m="138530">say</span> <span m="138680">you're</span> <span m="138850">trying</span> <span m="139160">to</span> <span m="139230">prove</span> <span m="139560">a</span> <span m="139760">equals</span> <span m="140060">b.</span> <span m="141600">Well,</span> <span m="141770">the</span> <span m="141880">proof</span> <span m="142350">sometimes</span> <span m="142690">will</span> <span m="142770">start</span> <span m="143170">with</span> <span m="143310">what</span> <span m="143530">you're</span> <span m="143630">trying</span> <span m="143920">to</span> <span m="143990">prove.</span> <span m="145080">And</span> <span m="145220">then</span> <span m="145380">they'll</span> <span m="145540">do</span> <span m="145680">a</span> <span m="145740">bunch</span> <span m="146010">of</span> <span m="146110">steps.</span> <span m="147200">And</span> <span m="147280">then</span> <span m="147400">you'll</span> <span m="147570">end</span> <span m="147750">up</span> <span m="147890">with</span> <span m="148060">1</span> <span m="148300">equals</span> <span m="148580">1,</span> <span m="149270">and</span> <span m="149545">you'll</span> <span m="149900">write</span> <span m="150090">a</span> <span m="150160">check,</span> <span m="152050">because</span> <span m="152520">you</span> <span m="152690">went</span> <span m="152900">from</span> <span m="153050">a</span> <span m="153180">equals</span> <span m="153430">b</span> <span m="153610">to 1</span> <span m="153880">equals</span> <span m="154150">1.</span> <span m="155440">And</span> <span m="155630">that's,</span> <span m="155850">of</span> <span m="155910">course,</span> <span m="156180">true.</span> </p>
<p><span m="157510">Now,</span> <span m="158880">that's</span> <span m="159140">not</span> <span m="159460">a</span> <span m="159520">good</span> <span m="159680">thing</span> <span m="159830">to</span> <span m="159900">do.</span> <span m="160250">It</span> <span m="160610">can</span> <span m="160950">be</span> <span m="161060">correct</span> <span m="161760">if</span> <span m="161960">the</span> <span m="162080">implications</span> <span m="162830">go</span> <span m="162980">this</span> <span m="163220">way.</span> <span m="165400">Because</span> <span m="165750">really,</span> <span m="166150">you're</span> <span m="166230">starting</span> <span m="166690">with</span> <span m="167860">a</span> <span m="167950">fact--</span> <span m="168350">1</span> <span m="168540">equals</span> <span m="168790">1--</span> <span m="169520">and</span> <span m="169810">deriving</span> <span m="170420">a</span> <span m="170550">equals</span> <span m="170810">b.</span> <span m="171160">So</span> <span m="171330">if,</span> <span m="171390">in</span> <span m="171530">fact,</span> <span m="172500">the</span> <span m="172600">implications</span> <span m="173260">work</span> <span m="173500">this</span> <span m="173730">way,</span> <span m="174000">then</span> <span m="174090">your</span> <span m="174210">proof</span> <span m="174500">is</span> <span m="174630">right.</span> </p>
<p><span m="175720">But</span> <span m="176110">most</span> <span m="176590">people,</span> <span m="177110">especially</span> <span m="177400">as</span> <span m="177460">you</span> <span m="177550">get</span> <span m="177720">farther</span> <span m="178020">along,</span> <span m="178930">think</span> <span m="179250">about</span> <span m="179520">it</span> <span m="179580">going</span> <span m="179840">from</span> <span m="179990">top</span> <span m="180320">to</span> <span m="180410">bottom.</span> <span m="182470">So</span> <span m="182650">don't</span> <span m="183160">use</span> <span m="183500">this</span> <span m="183680">technique,</span> <span m="184520">because</span> <span m="184720">it'll</span> <span m="184850">confuse</span> <span m="185430">people.</span> <span m="186330">You're</span> <span m="186450">liable</span> <span m="186740">to</span> <span m="186800">make</span> <span m="186980">a</span> <span m="187020">mistake.</span> <span m="187940">Just</span> <span m="188030">start</span> <span m="188450">with</span> <span m="188580">1</span> <span m="188790">equals</span> <span m="189030">1</span> <span m="189320">and</span> <span m="189770">work</span> <span m="190010">your</span> <span m="190120">way</span> <span m="190340">from</span> <span m="190510">there--</span> <span m="190790">top</span> <span m="191240">down</span> <span m="192400">for</span> <span m="192530">being</span> <span m="192780">in</span> <span m="192860">order.</span> </p>
<p><span m="196140">Now,</span> <span m="196300">good</span> <span m="196620">proofs</span> <span m="197040">are</span> <span m="197120">very</span> <span m="197490">much</span> <span m="197790">like</span> <span m="198330">good</span> <span m="198620">code.</span> <span m="201090">In</span> <span m="201190">fact,</span> <span m="201430">one</span> <span m="201550">of</span> <span m="201590">the</span> <span m="201690">reasons</span> <span m="202040">we</span> <span m="202180">care</span> <span m="202540">so</span> <span m="202730">much</span> <span m="203080">about</span> <span m="203350">teaching</span> <span m="203650">you</span> <span m="203750">how</span> <span m="203910">to</span> <span m="203980">write</span> <span m="204140">a</span> <span m="204200">good</span> <span m="204430">proof</span> <span m="205430">in</span> <span m="205540">computer</span> <span m="205920">science</span> <span m="206990">is</span> <span m="207190">so</span> <span m="207310">that</span> <span m="207580">later</span> <span m="207910">on,</span> <span m="209210">you'll</span> <span m="209370">be</span> <span m="209490">able</span> <span m="209630">to</span> <span m="209690">prove</span> <span m="210180">that your</span> <span m="210450">programs</span> <span m="211650">are</span> <span m="211790">doing</span> <span m="212360">what</span> <span m="212560">you</span> <span m="212700">expect--</span> <span m="213270">what</span> <span m="213430">they're</span> <span m="213550">supposed</span> <span m="213920">to</span> <span m="213990">do.</span> <span m="215680">Now,</span> <span m="215760">there</span> <span m="215990">are</span> <span m="216050">many</span> <span m="216710">famous</span> <span m="217130">examples</span> <span m="217810">where</span> <span m="218540">programs</span> <span m="219060">did</span> <span m="219240">not</span> <span m="219560">do</span> <span m="219720">what</span> <span m="219850">they</span> <span m="219980">were</span> <span m="220130">supposed</span> <span m="220430">to</span> <span m="220510">do,</span> <span m="220700">with</span> <span m="220850">disastrous</span> <span m="221550">consequences.</span> </p>
<p><span m="223340">The</span> <span m="223530">Airbus</span> <span m="223990">A300</span> <span m="225650">was</span> <span m="225830">one</span> <span m="225970">of</span> <span m="226010">the</span> <span m="226290">first</span> <span m="226790">commercial</span> <span m="227340">jets</span> <span m="228370">totally</span> <span m="228800">operated</span> <span m="229180">by</span> <span m="229290">software.</span> <span m="229720">It</span> <span m="230150">could</span> <span m="230310">take</span> <span m="230580">off,</span> <span m="230830">fly,</span> <span m="231210">and</span> <span m="231390">land</span> <span m="232090">totally</span> <span m="232460">by</span> <span m="232600">software.</span> <span m="234140">It was</span> <span m="234250">a</span> <span m="234310">major</span> <span m="234730">advance</span> <span m="235530">in</span> <span m="236140">the</span> <span m="236260">airline</span> <span m="236590">industry.</span> </p>
<p><span m="237600">The</span> <span m="237710">only</span> <span m="237880">problem</span> <span m="238330">was,</span> <span m="238740">that</span> <span m="238830">on</span> <span m="238990">one</span> <span m="239110">of</span> <span m="239150">the</span> <span m="239240">first</span> <span m="239580">flights</span> <span m="239880">of</span> <span m="239970">the</span> <span m="240110">A300,</span> <span m="241140">the</span> <span m="241260">software</span> <span m="241950">accidentally</span> <span m="242780">opened</span> <span m="243660">the</span> <span m="243800">rear</span> <span m="244100">door</span> <span m="244980">just</span> <span m="245240">before</span> <span m="245530">landing.</span> <span m="246580">And</span> <span m="246700">the</span> <span m="246770">plane</span> <span m="247020">crashed</span> <span m="247620">as</span> <span m="247800">a</span> <span m="247850">result.</span> <span m="248940">It was the</span> <span m="249040">first</span> <span m="249250">plane</span> <span m="249520">crash</span> <span m="249775">in</span> <span m="250030">commercial</span> <span m="250650">history</span> <span m="251500">because</span> <span m="251740">of</span> <span m="251800">a</span> <span m="251860">software</span> <span m="252300">bug.</span> </p>
<p><span m="254130">There's</span> <span m="254430">a</span> <span m="255220">famous</span> <span m="255570">radiation</span> <span m="256200">device</span> <span m="256660">for</span> <span m="256790">cancer</span> <span m="257170">patients</span> <span m="257589">called</span> <span m="257839">the</span> <span m="257920">Therac-25.</span> <span m="259970">It's</span> <span m="260130">famous</span> <span m="260630">because</span> <span m="262029">it</span> <span m="262110">got</span> <span m="262310">into</span> <span m="262480">a</span> <span m="262600">race</span> <span m="262920">condition</span> <span m="263350">occasionally,</span> <span m="264460">which</span> <span m="264690">caused</span> <span m="265010">the</span> <span m="265070">device</span> <span m="266070">to</span> <span m="266180">just</span> <span m="266710">slam</span> <span m="267290">the</span> <span m="267380">patient</span> <span m="267720">with</span> <span m="267810">radiation--</span> <span m="268510">so</span> <span m="268590">much</span> <span m="268820">so,</span> <span m="269310">it killed</span> <span m="269520">the</span> <span m="269600">patient.</span> <span m="270700">And</span> <span m="270800">they</span> <span m="270900">had</span> <span m="270910">multiple</span> <span m="271370">examples</span> <span m="271850">of</span> <span m="271920">this,</span> <span m="272280">and of</span> <span m="272590">course,</span> <span m="272960">a lot of</span> <span m="273210">lawsuits</span> <span m="273670">afterwards.</span> </p>
<p><span m="275490">How</span> <span m="275620">many</span> <span m="275810">of</span> <span m="275870">you</span> <span m="276020">all</span> <span m="276450">remember</span> <span m="276850">the</span> <span m="276990">2000</span> <span m="277540">election?</span> <span m="279830">A few of</span> <span m="280020">you</span> <span m="280160">do.</span> <span m="280430">OK,</span> <span m="280930">this</span> <span m="281360">will</span> <span m="281540">probably</span> <span m="281760">be</span> <span m="281830">the</span> <span m="281920">last</span> <span m="282380">class</span> <span m="282810">that</span> <span m="282920">remembers</span> <span m="283350">that.</span> <span m="284320">But</span> <span m="284380">that's</span> <span m="284590">where</span> <span m="284940">Al</span> <span m="285410">Gore</span> <span m="285650">was</span> <span m="285800">going</span> <span m="286000">up</span> <span m="286100">against</span> <span m="286390">George</span> <span m="286690">Bush--</span> <span m="287030">very</span> <span m="287560">close</span> <span m="287850">election--</span> <span m="288660">all</span> <span m="288900">came</span> <span m="289170">down</span> <span m="289390">to</span> <span m="289490">recounting</span> <span m="289930">the</span> <span m="290000">votes</span> <span m="290280">in</span> <span m="290370">Florida.</span> </p>
<p><span m="292060">But</span> <span m="292550">in</span> <span m="292660">fact,</span> <span m="293720">Al</span> <span m="293990">Gore</span> <span m="294310">got</span> <span m="294530">negative</span> <span m="295040">16,000</span> <span m="296030">votes</span> <span m="296370">in</span> <span m="296460">one</span> <span m="296700">county</span> <span m="298150">because</span> <span m="298600">of</span> <span m="298690">a</span> <span m="298750">software</span> <span m="299280">bug</span> <span m="299590">in</span> <span m="299660">the</span> <span m="299730">electronic</span> <span m="300570">voting</span> <span m="301630">booths,</span> <span m="301990">which</span> <span m="302150">I</span> <span m="302200">think</span> <span m="302440">they</span> <span m="302550">got</span> <span m="302740">rid</span> <span m="302900">of</span> <span m="302970">a</span> <span m="303040">lot</span> <span m="303320">of</span> <span m="303380">them</span> <span m="303930">in</span> <span m="303990">the</span> <span m="304220">election</span> <span m="304550">after</span> <span m="304790">that.</span> <span m="305000">Because</span> <span m="305380">the</span> <span m="305480">software</span> <span m="305900">was</span> <span m="306050">buggy.</span> <span m="307010">So</span> <span m="307220">poor</span> <span m="307500">Al</span> <span m="307710">had</span> <span m="307830">enough</span> <span m="308020">problem</span> <span m="308270">which</span> <span m="308530">chads</span> <span m="308920">and</span> <span m="309120">funny</span> <span m="309330">business</span> <span m="309650">in</span> <span m="309760">Florida.</span> <span m="310090">But getting</span> <span m="310440">negative</span> <span m="310840">16,000</span> <span m="311600">votes</span> <span m="312240">certainly</span> <span m="312580">didn't</span> <span m="312750">help</span> <span m="312950">his</span> <span m="313050">chances.</span> </p>
<p><span m="314890">Several</span> <span m="315200">years</span> <span m="315540">ago,</span> <span m="315900">a</span> <span m="316000">single</span> <span m="316350">faulty</span> <span m="316730">command</span> <span m="317820">in</span> <span m="317950">a</span> <span m="317990">computer</span> <span m="318430">system</span> <span m="318760">used</span> <span m="319070">by</span> <span m="319260">United</span> <span m="319740">and</span> <span m="319860">American</span> <span m="320250">Airlines</span> <span m="321320">grounded</span> <span m="321870">the</span> <span m="321990">entire</span> <span m="322510">fleets</span> <span m="322840">to</span> <span m="322970">both</span> <span m="323360">airlines</span> <span m="324420">for</span> <span m="324540">close</span> <span m="324800">to</span> <span m="324910">a</span> <span m="324980">day.</span> <span m="326210">Because</span> <span m="326450">they</span> <span m="326530">couldn't</span> <span m="326760">do</span> <span m="326910">anything.</span> <span m="328050">They're</span> <span m="328180">all</span> <span m="328310">run</span> <span m="328450">by</span> <span m="328550">computer,</span> <span m="328910">and</span> <span m="328980">the</span> <span m="329050">whole</span> <span m="329230">thing</span> <span m="329380">was</span> <span m="329500">screwed</span> <span m="329740">up.</span> <span m="331480">So</span> <span m="331500">there's</span> <span m="331670">lots</span> <span m="332170">more</span> <span m="332430">examples.</span> </p>
<p><span m="333400">We</span> <span m="333590">run</span> <span m="333790">into</span> <span m="333960">this</span> <span m="334200">issue</span> <span m="334550">all</span> <span m="334710">the</span> <span m="334810">time</span> <span m="335090">at</span> <span m="335160">Akamai.</span> <span m="336790">Akamai</span> <span m="337270">is</span> <span m="337440">a</span> <span m="337620">company</span> <span m="338010">started</span> <span m="339140">by</span> <span m="339370">MIT</span> <span m="339820">folks--</span> <span m="340120">by</span> <span m="340220">myself,</span> <span m="340760">and</span> <span m="340880">Danny</span> <span m="341140">Lewin,</span> <span m="341480">and</span> <span m="341760">a</span> <span m="341880">dozen</span> <span m="342220">undergrads</span> <span m="342870">in</span> <span m="342930">the</span> <span m="342990">late</span> <span m="343190">1990s.</span> <span m="344850">And</span> <span m="345930">we</span> <span m="346090">deliver</span> <span m="346950">a</span> <span m="347020">lot</span> <span m="347260">of</span> <span m="347300">the</span> <span m="347370">content</span> <span m="347720">you</span> <span m="347800">get</span> <span m="347900">on</span> <span m="347970">the</span> <span m="348060">web--</span> <span m="348340">Facebook,</span> <span m="349950">all</span> <span m="350120">the</span> <span m="350220">search</span> <span m="350530">engines.</span> <span m="351850">A</span> <span m="351980">lot</span> <span m="352230">of</span> <span m="352270">the</span> <span m="352330">stuff</span> <span m="352530">you</span> <span m="352660">go</span> <span m="352890">to</span> <span m="353440">comes</span> <span m="353680">from</span> <span m="353790">our</span> <span m="353910">servers.</span> </p>
<p><span m="354790">So</span> <span m="354910">we've</span> <span m="355000">got</span> <span m="355080">to</span> <span m="355120">be</span> <span m="355220">very</span> <span m="355540">careful</span> <span m="356470">that</span> <span m="356600">we</span> <span m="356710">don't</span> <span m="356880">have</span> <span m="357060">software</span> <span m="357510">bugs.</span> <span m="358050">Now, in</span> <span m="358240">fact</span> <span m="358500">we</span> <span m="358580">do.</span> <span m="358830">And</span> <span m="358950">we</span> <span m="359260">catch them</span> <span m="359590">every</span> <span m="359740">once</span> <span m="359960">in</span> <span m="360000">awhile.</span> </p>
<p><span m="360710">But</span> <span m="360850">if</span> <span m="360920">we</span> <span m="361010">got</span> <span m="361160">a</span> <span m="361210">bad</span> <span m="361520">one,</span> <span m="361910">it</span> <span m="362080">would</span> <span m="362250">bring</span> <span m="362510">down</span> <span m="362740">all</span> <span m="362890">the</span> <span m="362980">sites</span> <span m="363190">you</span> <span m="363290">go</span> <span m="363470">to</span> <span m="363610">you.</span> <span m="363760">You wouldn't</span> <span m="363850">be</span> <span m="363920">able</span> <span m="364040">to</span> <span m="364100">go</span> <span m="364250">to</span> <span m="364300">those</span> <span m="364500">sites</span> <span m="364780">anymore.</span> <span m="366000">And</span> <span m="366090">everybody</span> <span m="366410">would</span> <span m="366540">notice.</span> <span m="366980">It would be</span> <span m="367120">sort</span> <span m="367330">of</span> <span m="367600">embarrassing.</span> </p>
<p><span m="370740">Now,</span> <span m="372340">this</span> <span m="372520">really</span> <span m="372940">does</span> <span m="373270">matter.</span> <span m="374680">And</span> <span m="375400">this</span> <span m="375550">is</span> <span m="375650">going</span> <span m="375750">to</span> <span m="375800">sound</span> <span m="376070">a</span> <span m="376120">little</span> <span m="376390">scary,</span> <span m="377310">but</span> <span m="377490">someday--</span> <span m="378380">probably</span> <span m="379020">30,</span> <span m="379440">20</span> <span m="379720">years from</span> <span m="380030">now,</span> <span m="380280">somewhere</span> <span m="380620">in</span> <span m="380690">there--</span> <span m="382050">it's</span> <span m="382400">possible</span> <span m="382770">that</span> <span m="382870">all</span> <span m="383130">of</span> <span m="383240">us,</span> <span m="384460">our</span> <span m="384640">lives</span> <span m="385080">may</span> <span m="385230">depend</span> <span m="385670">on</span> <span m="386460">the</span> <span m="386560">software</span> <span m="387250">that</span> <span m="387550">some</span> <span m="387710">of</span> <span m="387800">you</span> <span m="387970">write.</span> </p>
<p><span m="389490">In</span> <span m="389540">fact,</span> <span m="389740">to</span> <span m="389820">bring</span> <span m="390190">this</span> <span m="390370">home</span> <span m="390660">how</span> <span m="390800">scary</span> <span m="391190">it</span> <span m="391250">is,</span> <span m="392010">look</span> <span m="392220">at</span> <span m="392260">the</span> <span m="392350">person</span> <span m="392680">sitting</span> <span m="392940">next</span> <span m="393200">to</span> <span m="393260">you.</span> <span m="394550">And</span> <span m="394730">imagine</span> <span m="395490">that</span> <span m="395590">in</span> <span m="395700">25</span> <span m="396280">years,</span> <span m="396650">your</span> <span m="396880">life</span> <span m="397550">depends</span> <span m="398080">on</span> <span m="398750">whether</span> <span m="399010">their</span> <span m="399340">code</span> <span m="399830">does</span> <span m="400060">what</span> <span m="400180">it's</span> <span m="400280">supposed</span> <span m="400580">to</span> <span m="400650">do--</span> <span m="402200">little</span> <span m="402420">scary.</span> <span m="407390">That's</span> <span m="407830">why</span> <span m="408380">we</span> <span m="408640">are</span> <span m="408730">very</span> <span m="409240">motivated</span> <span m="410980">to</span> <span m="411090">help</span> <span m="411340">you</span> <span m="411530">learn</span> <span m="411900">how</span> <span m="412080">to</span> <span m="412160">make</span> <span m="412460">rock-solid</span> <span m="413590">arguments,</span> <span m="415660">so</span> <span m="415750">you</span> <span m="415900">don't</span> <span m="416150">have</span> <span m="416390">code</span> <span m="416710">that</span> <span m="416840">fries</span> <span m="417370">one</span> <span m="417530">of</span> <span m="417600">your</span> <span m="417730">classmates</span> <span m="418290">someday,</span> <span m="418810">or</span> <span m="419630">puts</span> <span m="419850">him</span> <span m="419990">in</span> <span m="420380">a</span> <span m="420560">bad</span> <span m="420920">plane</span> <span m="421200">situation.</span> </p>
<p><span m="423040">Now</span> <span m="423240">unfortunately,</span> <span m="424590">writing</span> <span m="425280">rock-solid</span> <span m="426100">proofs</span> <span m="427100">is</span> <span m="427290">a</span> <span m="427350">very</span> <span m="427680">hard</span> <span m="427940">thing</span> <span m="428140">to</span> <span m="428220">do.</span> <span m="428910">Even</span> <span m="429210">the</span> <span m="429300">world's</span> <span m="429710">best</span> <span m="430130">mathematicians</span> <span m="431380">mess</span> <span m="431690">them</span> <span m="431850">up</span> <span m="432400">on</span> <span m="432580">a</span> <span m="432660">regular</span> <span m="433030">basis.</span> <span m="433550">In</span> <span m="433650">fact,</span> <span m="433870">it's</span> <span m="433930">estimated</span> <span m="435020">that</span> <span m="435180">one</span> <span m="435760">third</span> <span m="436560">of</span> <span m="436720">all</span> <span m="437070">published</span> <span m="437540">proofs</span> <span m="438570">have</span> <span m="438770">bugs,</span> <span m="439540">have</span> <span m="439760">flaws,</span> <span m="440570">that</span> <span m="440710">render</span> <span m="440930">the</span> <span m="441030">proof</span> <span m="441470">incorrect.</span> </p>
<p><span m="443330">The</span> <span m="443460">trouble</span> <span m="443930">often</span> <span m="444300">arises</span> <span m="444860">because</span> <span m="445560">we</span> <span m="445680">get</span> <span m="445840">lazy.</span> <span m="448040">We</span> <span m="448180">don't</span> <span m="448540">write</span> <span m="448790">down</span> <span m="449000">all</span> <span m="449130">the</span> <span m="449240">details</span> <span m="449740">or</span> <span m="449820">all</span> <span m="450020">the</span> <span m="450110">steps.</span> <span m="450570">Because,</span> <span m="451040">wow,</span> <span m="451630">it's</span> <span m="452010">clear.</span> <span m="452880">Let's</span> <span m="453070">just</span> <span m="453270">move</span> <span m="453460">on.</span> </p>
<p><span m="455740">Now,</span> <span m="457200">this</span> <span m="457480">can</span> <span m="457850">be</span> <span m="458010">OK.</span> <span m="459330">Not such</span> <span m="459640">a</span> <span m="459680">good</span> <span m="459850">practice,</span> <span m="460360">but</span> <span m="460550">it</span> <span m="461120">dramatically</span> <span m="462010">increases</span> <span m="462470">the</span> <span m="462570">chances</span> <span m="463110">of</span> <span m="463200">making</span> <span m="463510">a</span> <span m="463550">mistake.</span> <span m="465100">And</span> <span m="465300">there's</span> <span m="465420">some</span> <span m="465540">very</span> <span m="465810">famous</span> <span m="466160">examples</span> <span m="466650">in the</span> <span m="466750">math</span> <span m="467030">literature</span> <span m="467670">from</span> <span m="467820">the</span> <span m="467910">world's</span> <span m="468190">most</span> <span m="468400">famous</span> <span m="468730">mathematicians.</span> </p>
<p><span m="470320">Gauss--</span> <span m="470780">and</span> <span m="470880">we'll</span> <span m="471280">talk</span> <span m="471510">about</span> <span m="471730">Gauss</span> <span m="472340">later</span> <span m="472610">during</span> <span m="472810">the</span> <span m="472900">term--</span> <span m="473160">he's</span> <span m="473250">one</span> <span m="473380">of</span> <span m="473420">the</span> <span m="473460">most</span> <span m="473670">famous</span> <span m="474000">mathematicians</span> <span m="474620">ever.</span> <span m="475420">He</span> <span m="475600">wrote</span> <span m="475780">his</span> <span m="475900">PhD</span> <span m="476490">thesis</span> <span m="476910">in</span> <span m="477010">1799.</span> <span m="479640">And</span> <span m="479900">it's</span> <span m="480340">usually</span> <span m="480850">referred</span> <span m="481320">to</span> <span m="481470">as</span> <span m="481580">the</span> <span m="481680">first</span> <span m="482020">rigorous</span> <span m="482610">proof</span> <span m="483440">of</span> <span m="483580">the</span> <span m="483660">fundamental</span> <span m="484200">theorem</span> <span m="484510">of</span> <span m="484620">algebra.</span> </p>
<p><span m="485350">And</span> <span m="485610">that</span> <span m="485820">says</span> <span m="486360">that</span> <span m="486480">every</span> <span m="486810">polynomial</span> <span m="488130">has</span> <span m="488350">a</span> <span m="488400">0</span> <span m="488850">over</span> <span m="489000">the</span> <span m="489100">complex</span> <span m="489580">numbers--</span> <span m="490130">something</span> <span m="490410">probably</span> <span m="490810">a</span> <span m="490870">lot</span> <span m="491040">of</span> <span m="491100">you</span> <span m="491200">learned</span> <span m="491400">in</span> <span m="491460">high</span> <span m="491650">school.</span> <span m="492460">You</span> <span m="492530">get</span> <span m="492620">a</span> <span m="492680">polynomial.</span> <span m="493200">You</span> <span m="493250">can</span> <span m="493400">find</span> <span m="493700">roots</span> <span m="493960">of the</span> <span m="494060">polynomial</span> <span m="494380">over</span> <span m="494700">the</span> <span m="494830">complex</span> <span m="495340">numbers.</span> </p>
<p><span m="497100">But</span> <span m="497250">his</span> <span m="497450">thesis</span> <span m="497810">contains</span> <span m="498370">the</span> <span m="498460">following</span> <span m="499000">quote,</span> <span m="500580">"If</span> <span m="500750">a</span> <span m="500820">branch</span> <span m="501430">of</span> <span m="501520">an</span> <span m="501610">algebraic</span> <span m="502270">curve</span> <span m="502930">enters</span> <span m="503320">a</span> <span m="503390">bounded</span> <span m="503860">region,</span> <span m="504730">it</span> <span m="504910">must</span> <span m="505200">necessarily</span> <span m="506040">leave</span> <span m="506330">again.</span> <span m="507870">Nobody,</span> <span m="508470">to</span> <span m="508590">my</span> <span m="508780">knowledge,</span> <span m="509260">has</span> <span m="509480">ever</span> <span m="509780">doubted</span> <span m="510180">this</span> <span m="510390">fact."</span> <span m="511480">Warning</span> <span m="511870">signs--</span> <span m="512419">buzzers--</span> <span m="512880">should</span> <span m="513039">be</span> <span m="513150">going</span> <span m="513429">off</span> <span m="513600">in</span> <span m="513659">your</span> <span m="513770">brain.</span> </p>
<p><span m="514679">"But</span> <span m="514789">if</span> <span m="514870">anybody</span> <span m="515299">desires</span> <span m="515850">it,</span> <span m="516000">then</span> <span m="516159">on</span> <span m="516299">another</span> <span m="516620">occasion,</span> <span m="517090">I</span> <span m="517190">intend</span> <span m="517669">to</span> <span m="517710">give</span> <span m="517890">a</span> <span m="517960">demonstration</span> <span m="518710">which</span> <span m="518880">will</span> <span m="519030">leave</span> <span m="519250">no</span> <span m="519419">doubt."</span> <span m="521100">So</span> <span m="521220">he's</span> <span m="521490">using</span> <span m="521890">something</span> <span m="522280">that he</span> <span m="522460">believes</span> <span m="522760">to</span> <span m="522820">be</span> <span m="522940">true.</span> <span m="524620">He sort</span> <span m="524750">of</span> <span m="525020">thinks</span> <span m="525250">it's</span> <span m="525390">clear,</span> <span m="526490">that</span> <span m="526600">everybody</span> <span m="527000">knows</span> <span m="527320">it.</span> <span m="528420">But</span> <span m="528540">he</span> <span m="528660">writes</span> <span m="528910">this</span> <span m="529230">down.</span> </p>
<p><span m="529530">And</span> <span m="529590">when</span> <span m="529710">you're</span> <span m="529830">writing</span> <span m="530070">that</span> <span m="530250">down,</span> <span m="530440">you</span> <span m="530530">know</span> <span m="531120">there's</span> <span m="531290">a</span> <span m="531350">problem.</span> <span m="532730">And</span> <span m="532870">in</span> <span m="532950">fact,</span> <span m="533320">there's</span> <span m="533700">another</span> <span m="533980">Fields</span> <span m="534360">medalist--</span> <span m="534700">we</span> <span m="534810">talked</span> <span m="535090">about</span> <span m="535470">Fields</span> <span m="535710">medals</span> <span m="536050">last</span> <span m="536330">week--</span> <span m="537020">Stephen</span> <span m="537310">Smale</span> <span m="537800">writes</span> <span m="539090">that</span> <span m="539620">this</span> <span m="539840">was</span> <span m="539990">an</span> <span m="540080">immense</span> <span m="540660">gap</span> <span m="541260">in</span> <span m="541410">the</span> <span m="541510">proof</span> <span m="542010">that</span> <span m="542120">was</span> <span m="542300">not</span> <span m="542650">filled</span> <span m="543000">until</span> <span m="543270">1920--</span> <span m="544120">more</span> <span m="544545">than</span> <span m="544970">100</span> <span m="545370">years</span> <span m="545630">later.</span> <span m="546500">So</span> <span m="546660">Gauss</span> <span m="546960">never</span> <span m="547170">could</span> <span m="547400">give</span> <span m="547540">the</span> <span m="547630">proof,</span> <span m="548310">and</span> <span m="548450">nobody</span> <span m="548750">did</span> <span m="548970">for</span> <span m="549340">over</span> <span m="549580">a</span> <span m="549650">century</span> <span m="550030">more.</span> </p>
<p><span m="551870">Remember</span> <span m="552180">the</span> <span m="552530">Poincare</span> <span m="552820">conjecture</span> <span m="553520">from</span> <span m="553670">last</span> <span m="553940">week?</span> <span m="554830">We</span> <span m="554890">had</span> <span m="555050">Colbert</span> <span m="555450">talking</span> <span m="555730">about</span> <span m="555960">that.</span> <span m="557210">In</span> <span m="557390">1900,</span> <span m="558440">Poincare</span> <span m="559480">claimed</span> <span m="560030">it</span> <span m="560090">was</span> <span m="560230">a</span> <span m="560290">simple</span> <span m="560700">fact.</span> <span m="562440">Four</span> <span m="562740">years</span> <span m="562980">later,</span> <span m="563580">he</span> <span m="563870">decided</span> <span m="564290">it</span> <span m="564350">wasn't</span> <span m="564720">so</span> <span m="564860">simple,</span> <span m="565790">and he</span> <span m="566030">demoted</span> <span m="566460">his</span> <span m="566590">claim</span> <span m="566870">to</span> <span m="566930">the</span> <span m="567040">status</span> <span m="567480">of</span> <span m="567570">a</span> <span m="567620">conjecture.</span> <span m="568600">And</span> <span m="568710">of</span> <span m="568790">course,</span> <span m="569060">this</span> <span m="569230">became</span> <span m="569690">the</span> <span m="569770">famous</span> <span m="570610">Poincare</span> <span m="571110">conjecture,</span> <span m="572240">which</span> <span m="572450">took</span> <span m="572620">another</span> <span m="572960">century</span> <span m="573460">to</span> <span m="573570">solve.</span> </p>
<p><span m="575480">So</span> <span m="575580">when</span> <span m="575670">you</span> <span m="576570">think</span> <span m="576840">you</span> <span m="576930">see</span> <span m="577140">yourself</span> <span m="577520">doing</span> <span m="577780">this,</span> <span m="578380">famous</span> <span m="578710">mathematicians</span> <span m="579320">do</span> <span m="579470">it</span> <span m="579560">too,</span> <span m="579760">and</span> <span m="579850">it</span> <span m="580100">catches</span> <span m="580540">them.</span> <span m="581860">So</span> <span m="581990">you</span> <span m="582080">want</span> <span m="582280">to</span> <span m="582320">not</span> <span m="582560">try</span> <span m="582690">to</span> <span m="582760">do</span> <span m="582870">that.</span> <span m="583130">Now,</span> <span m="583320">in</span> <span m="583420">fact,</span> <span m="583690">just</span> <span m="583940">to</span> <span m="584060">really</span> <span m="584380">bring</span> <span m="584600">home</span> <span m="585330">the</span> <span m="585450">point,</span> <span m="586260">we've</span> <span m="586490">made</span> <span m="586810">a</span> <span m="587650">top</span> <span m="587950">10</span> <span m="588180">list</span> <span m="588970">of proof</span> <span m="589500">techniques</span> <span m="589940">you</span> <span m="590040">should</span> <span m="590280">not</span> <span m="590630">use</span> <span m="591580">in</span> <span m="591730">6.042.</span> </p>
<p><span m="592330">So</span> <span m="592550">everybody</span> <span m="592880">go</span> <span m="593040">to</span> <span m="593100">the</span> <span m="593180">handout</span> <span m="593660">here.</span> <span m="594940">In fact,</span> <span m="595090">there's</span> <span m="595140">lot</span> <span m="595470">of</span> <span m="595540">them</span> <span m="595940">on</span> <span m="596090">the</span> <span m="596170">back,</span> <span m="596460">too, but</span> <span m="596620">on the</span> <span m="596810">front</span> <span m="597090">page,</span> <span m="597370">there's</span> <span m="597560">the</span> <span m="597660">top</span> <span m="597920">10.</span> <span m="598730">So</span> <span m="598840">we're</span> <span m="598930">going</span> <span m="599020">to</span> <span m="599080">go</span> <span m="599310">through</span> <span m="599490">these.</span> <span m="601220">These are</span> <span m="601260">the</span> <span m="601560">ones</span> <span m="601950">we've</span> <span m="602090">most</span> <span m="602440">observed</span> <span m="602910">in</span> <span m="602980">6.042</span> <span m="603770">over</span> <span m="603950">the</span> <span m="604070">years.</span> </p>
<p><span m="605020">Number</span> <span m="605330">10--</span> <span m="606460">proof</span> <span m="606770">by</span> <span m="606920">throwing</span> <span m="607340">in</span> <span m="607410">the</span> <span m="607510">kitchen</span> <span m="607870">sink.</span> <span m="609270">The</span> <span m="609400">author</span> <span m="609730">writes</span> <span m="610040">down</span> <span m="610320">every</span> <span m="610650">theorem</span> <span m="611020">known</span> <span m="611260">to</span> <span m="611330">mankind,</span> <span m="611930">and</span> <span m="612010">then</span> <span m="612160">adds</span> <span m="612390">a</span> <span m="612460">few</span> <span m="612680">more</span> <span m="613560">for</span> <span m="613690">good</span> <span m="613890">measure.</span> <span m="614930">Now,</span> <span m="615020">this</span> <span m="615210">is</span> <span m="615340">good,</span> <span m="615520">because</span> <span m="615700">when</span> <span m="615820">you're</span> <span m="615940">questioned</span> <span m="616430">later--</span> <span m="616820">you're</span> <span m="616990">trying</span> <span m="617200">to</span> <span m="617250">get</span> <span m="617390">a</span> <span m="617430">better</span> <span m="617720">score</span> <span m="618010">on</span> <span m="618110">your</span> <span m="618230">exam--</span> <span m="619780">you</span> <span m="619830">say,</span> <span m="620300">look,</span> <span m="620520">the</span> <span m="620620">proof</span> <span m="620900">contains</span> <span m="621320">all</span> <span m="621480">the</span> <span m="621570">key</span> <span m="621750">facts.</span> <span m="622810">They're</span> <span m="623000">all</span> <span m="623150">here.</span> </p>
<p><span m="625010">And</span> <span m="625180">so</span> <span m="625290">that</span> <span m="625570">does</span> <span m="625810">help.</span> <span m="626030">You</span> <span m="626120">get</span> <span m="626250">extra</span> <span m="626460">credit</span> <span m="626680">if</span> <span m="626790">all</span> <span m="626890">the</span> <span m="626970">facts</span> <span m="627290">are</span> <span m="627320">there.</span> <span m="627670">We</span> <span m="627880">are</span> <span m="627910">literally</span> <span m="628330">seen</span> <span m="629470">students</span> <span m="630020">copy</span> <span m="630380">over</span> <span m="630540">their</span> <span m="630670">crib</span> <span m="630980">sheet,</span> <span m="631290">if</span> <span m="631370">they</span> <span m="631470">got</span> <span m="631620">enough</span> <span m="631790">time,</span> <span m="632190">for a</span> <span m="632350">problem</span> <span m="632630">they're</span> <span m="632730">not</span> <span m="633350">knowing</span> <span m="633600">how</span> <span m="633760">to</span> <span m="633860">solve,</span> <span m="634630">just</span> <span m="634850">to</span> <span m="634920">get</span> <span m="635050">all</span> <span m="635290">of</span> <span m="635430">the</span> <span m="635530">facts</span> <span m="635870">in</span> <span m="635960">there.</span> </p>
<p><span m="637250">Number</span> <span m="637400">nine--</span> <span m="637810">proof</span> <span m="638090">by</span> <span m="638260">example--</span> <span m="639340">the</span> <span m="639490">author</span> <span m="639760">gives</span> <span m="640060">the</span> <span m="640160">case</span> <span m="640460">n</span> <span m="640590">equals</span> <span m="640850">2 and</span> <span m="641230">suggests</span> <span m="641720">it</span> <span m="641780">contains</span> <span m="642290">most</span> <span m="642600">of</span> <span m="642640">the</span> <span m="642730">ideas</span> <span m="643050">of</span> <span m="643090">the</span> <span m="643150">general</span> <span m="643480">proof.</span> <span m="644480">In fact,</span> <span m="644620">a</span> <span m="644690">student</span> <span m="645030">before a</span> <span m="645210">class</span> <span m="645540">was</span> <span m="645650">telling</span> <span m="645910">me</span> <span m="646150">on one</span> <span m="646280">of</span> <span m="646320">the</span> <span m="646410">homework</span> <span m="646740">problems,</span> <span m="647690">he</span> <span m="647790">talked</span> <span m="648050">about</span> <span m="648290">it to</span> <span m="648390">his</span> <span m="648490">non-math</span> <span m="649010">friends,</span> <span m="650410">and</span> <span m="650550">they</span> <span m="650660">gave</span> <span m="650930">examples,</span> <span m="652390">not</span> <span m="652640">the</span> <span m="652710">proof.</span> </p>
<p><span m="654220">Number</span> <span m="654590">eight--</span> <span m="654810">proof</span> <span m="655080">by</span> <span m="655210">vigorous</span> <span m="655640">hand-waving--</span> <span m="657090">one</span> <span m="657210">of</span> <span m="657270">my</span> <span m="657490">favorites--</span> <span m="657930">I</span> <span m="657990">get</span> <span m="658110">up</span> <span m="658260">here</span> <span m="658430">and</span> <span m="658520">wave</span> <span m="658760">my</span> <span m="658930">hands.</span> <span m="659440">And</span> <span m="659790">it</span> <span m="659910">must</span> <span m="660160">be</span> <span m="660300">true.</span> </p>
<p><span m="663330">Number</span> <span m="663620">seven--</span> <span m="664130">proof</span> <span m="664550">by</span> <span m="664680">cumbersome</span> <span m="665280">notation--</span> <span m="666890">here</span> <span m="667110">the</span> <span m="667250">reader</span> <span m="667510">gets</span> <span m="667730">hopelessly</span> <span m="668230">confused,</span> <span m="669080">gives</span> <span m="669400">up,</span> <span m="669590">and</span> <span m="669710">says</span> <span m="669940">OK.</span> <span m="671060">In</span> <span m="671160">fact,</span> <span m="671380">I</span> <span m="671470">once</span> <span m="671730">had</span> <span m="671860">a</span> <span m="671900">grad</span> <span m="672210">student,</span> <span m="673130">and</span> <span m="673340">we</span> <span m="673450">called</span> <span m="673730">him</span> <span m="673880">the</span> <span m="674040">encryptor,</span> <span m="675340">because</span> <span m="675610">he</span> <span m="675740">could</span> <span m="675890">take</span> <span m="676120">the</span> <span m="676220">simplest</span> <span m="676870">proof</span> <span m="677800">and</span> <span m="678040">so</span> <span m="678430">encrypt</span> <span m="678775">it</span> <span m="679120">in</span> <span m="679250">notation</span> <span m="679900">and</span> <span m="680230">God</span> <span m="680420">knows</span> <span m="680620">what,</span> <span m="681290">you</span> <span m="681440">could</span> <span m="681590">never</span> <span m="681920">understand</span> <span m="682380">the</span> <span m="682450">proof</span> <span m="683170">and</span> <span m="683390">figure</span> <span m="683650">out</span> <span m="683740">if</span> <span m="683820">it</span> <span m="683900">was</span> <span m="684030">really</span> <span m="684350">right</span> <span m="684560">or</span> <span m="684640">wrong.</span> </p>
<p><span m="685990">That</span> <span m="686170">goes</span> <span m="686370">well</span> <span m="686660">with</span> <span m="686760">number</span> <span m="687040">six--</span> <span m="687440">proof</span> <span m="687690">by</span> <span m="687880">exhaustion--</span> <span m="689520">so</span> <span m="689730">does</span> <span m="689920">throwing</span> <span m="690150">in</span> <span m="690220">the</span> <span m="690300">kitchen</span> <span m="690610">sink.</span> <span m="691850">Number</span> <span m="692060">five--</span> <span m="692520">proof</span> <span m="692920">by</span> <span m="693110">omission--</span> <span m="693870">you</span> <span m="694050">will</span> <span m="694160">see</span> <span m="694330">that</span> <span m="694500">a</span> <span m="694560">lot,</span> <span m="695560">also</span> <span m="695860">by</span> <span m="696100">faculty--</span> <span m="697270">"The</span> <span m="697390">reader</span> <span m="697680">may</span> <span m="697850">easily</span> <span m="698260">supply</span> <span m="698650">the</span> <span m="698770">details.</span> <span m="700300">The</span> <span m="700460">other</span> <span m="700650">cases</span> <span m="701060">are</span> <span m="701150">analogous--</span> <span m="702160">trivial."</span> </p>
<p><span m="703190">Experts</span> <span m="703590">use</span> <span m="703760">that</span> <span m="703910">all</span> <span m="704030">the</span> <span m="704100">time.</span> <span m="704400">You</span> <span m="704510">can</span> <span m="704670">find</span> <span m="704840">a</span> <span m="704880">lot</span> <span m="705030">of</span> <span m="705080">my</span> <span m="705220">papers</span> <span m="705600">online.</span> <span m="706340">"The</span> <span m="706440">proof</span> <span m="706820">is</span> <span m="706980">trivial"--</span> <span m="707860">very</span> <span m="708070">bad.</span> <span m="708840">Every</span> <span m="709030">once</span> <span m="709280">in</span> <span m="709320">a while,</span> <span m="709670">it</span> <span m="709760">turned</span> <span m="709950">out</span> <span m="710100">not</span> <span m="710280">to</span> <span m="710340">be</span> <span m="710420">so</span> <span m="710580">trivial.</span> </p>
<p><span m="711920">Number</span> <span m="712190">four--</span> <span m="712620">proof</span> <span m="712960">by</span> <span m="713130">picture--</span> <span m="714090">we've</span> <span m="714310">seen</span> <span m="714620">those,</span> <span m="715170">saw</span> <span m="715420">one on</span> <span m="715620">Tuesday.</span> <span m="717000">Number</span> <span m="717270">three--</span> <span m="717710">proof</span> <span m="718060">by</span> <span m="718230">vehement</span> <span m="719410">assertion--</span> <span m="720030">sort</span> <span m="720230">of</span> <span m="720320">like</span> <span m="720560">the</span> <span m="720650">hand-waving.</span> <span m="721800">The</span> <span m="721910">more</span> <span m="722100">forceful</span> <span m="722770">you</span> <span m="722900">are</span> <span m="723130">in your</span> <span m="723310">argument,</span> <span m="723780">the</span> <span m="723870">higher</span> <span m="724220">you</span> <span m="724360">raise</span> <span m="724610">your</span> <span m="724740">voice,</span> <span m="725590">the</span> <span m="725670">more</span> <span m="725840">intimidating</span> <span m="726400">you</span> <span m="726490">become,</span> <span m="727900">doesn't</span> <span m="728160">make</span> <span m="728350">it</span> <span m="728450">more</span> <span m="728630">true.</span> </p>
<p><span m="730160">Number</span> <span m="730400">two--</span> <span m="730680">proof</span> <span m="731060">by</span> <span m="731230">appeal</span> <span m="731640">to</span> <span m="731850">intuition--</span> <span m="733570">you</span> <span m="733870">say,</span> <span m="734350">"any</span> <span m="734650">moron</span> <span m="735210">knows</span> <span m="735550">that."</span> <span m="737230">Well,</span> <span m="737620">you're</span> <span m="737750">sort</span> <span m="737920">of</span> <span m="738000">reluctant</span> <span m="738530">to</span> <span m="738650">now</span> <span m="739000">question</span> <span m="739400">it.</span> <span m="740480">And</span> <span m="740650">then</span> <span m="740740">number</span> <span m="741050">one--</span> <span m="741380">proof</span> <span m="741670">by</span> <span m="741800">reference</span> <span m="742340">to</span> <span m="742450">eminent</span> <span m="742780">authority.</span> <span m="743430">I</span> <span m="743590">saw</span> <span m="743830">Fermat</span> <span m="744090">on</span> <span m="744350">the</span> <span m="744410">elevator</span> <span m="744840">and</span> <span m="744900">he</span> <span m="744980">said</span> <span m="745210">he</span> <span m="745310">had</span> <span m="745470">a proof.</span> </p>
<p><span m="746480">I</span> <span m="746670">don't</span> <span m="747030">think</span> <span m="747250">so.</span> <span m="748330">Fermat's</span> <span m="748910">been</span> <span m="749110">dead</span> <span m="749460">for</span> <span m="749600">over</span> <span m="749800">300</span> <span m="750210">years.</span> <span m="751420">And</span> <span m="751540">he</span> <span m="751640">was</span> <span m="751780">not</span> <span m="752070">so</span> <span m="752250">reliable</span> <span m="752790">in</span> <span m="752830">the</span> <span m="752910">first</span> <span m="753220">place.</span> <span m="754840">In</span> <span m="754980">fact,</span> <span m="755320">he has</span> <span m="755590">one</span> <span m="755740">of</span> <span m="755790">the</span> <span m="755850">most</span> <span m="756090">famous</span> <span m="756830">assertions</span> <span m="757940">that</span> <span m="758060">turned</span> <span m="758360">into</span> <span m="758620">a</span> <span m="758680">conjecture,</span> <span m="759270">called</span> <span m="759830">Fermat's</span> <span m="760160">last</span> <span m="760630">theorem.</span> <span m="761860">And it</span> <span m="762050">was</span> <span m="762210">not</span> <span m="762490">really</span> <span m="762760">a</span> <span m="762820">theorem</span> <span m="763560">by</span> <span m="763660">the</span> <span m="763770">time</span> <span m="763970">he</span> <span m="764100">died.</span> </p>
<p><span m="767620">How</span> <span m="767760">many</span> <span m="767950">people</span> <span m="768220">have</span> <span m="768310">heard</span> <span m="768590">of</span> <span m="768680">Fermat's</span> <span m="769180">last</span> <span m="769460">theorem?</span> <span m="769860">Raise</span> <span m="770060">your</span> <span m="770170">hand.</span> <span m="771010">Yeah,</span> <span m="771250">pretty</span> <span m="771470">famous.</span> <span m="772920">It's</span> <span m="773140">the</span> <span m="773240">one</span> <span m="773500">theorem</span> <span m="773980">he</span> <span m="774100">proved</span> <span m="774560">that</span> <span m="775390">actually, he</span> <span m="775810">didn't</span> <span m="776020">prove.</span> <span m="776935">It</span> <span m="777330">says,</span> <span m="777690">for</span> <span m="777860">all</span> <span m="778140">n</span> <span m="778530">bigger</span> <span m="778790">than</span> <span m="778970">2,</span> <span m="779850">there</span> <span m="779990">does</span> <span m="780190">not</span> <span m="780620">exist</span> <span m="782530">an</span> <span m="782770">x,</span> <span m="783170">y,</span> <span m="783970">and</span> <span m="784160">z</span> <span m="784990">in</span> <span m="785100">the</span> <span m="785190">natural</span> <span m="785590">numbers</span> <span m="788220">plus</span> <span m="788640">the</span> <span m="788730">positive</span> <span m="789220">natural</span> <span m="789560">numbers</span> <span m="790400">such</span> <span m="790700">that</span> <span m="791230">x</span> <span m="791480">to</span> <span m="791560">the</span> <span m="791710">n</span> <span m="792040">plus</span> <span m="792370">y</span> <span m="792600">to</span> <span m="792680">the</span> <span m="792840">n</span> <span m="793540">equals</span> <span m="794000">z</span> <span m="794110">to</span> <span m="794190">the</span> <span m="794360">n.</span> </p>
<p><span m="795540">Now</span> <span m="795650">of</span> <span m="795720">course,</span> <span m="795990">for</span> <span m="796110">n</span> <span m="796390">equals</span> <span m="796830">2,</span> <span m="798150">that's</span> <span m="798370">just</span> <span m="798530">the</span> <span m="798620">Pythagorean</span> <span m="799220">theorem.</span> <span m="799580">You</span> <span m="799700">can</span> <span m="799840">find</span> <span m="800240">three,</span> <span m="801210">four,</span> <span m="801560">and</span> <span m="801660">five,</span> <span m="802090">when</span> <span m="802180">you</span> <span m="802240">square</span> <span m="802650">them</span> <span m="803380">satisfy</span> <span m="803880">that.</span> <span m="804860">But</span> <span m="804870">it</span> <span m="804980">says</span> <span m="805460">this</span> <span m="805650">does</span> <span m="805830">not</span> <span m="806240">work</span> <span m="806580">for</span> <span m="806830">any</span> <span m="807100">higher</span> <span m="808570">value,</span> <span m="808970">any</span> <span m="809150">higher</span> <span m="809440">power,</span> <span m="809700">than</span> <span m="809960">2.</span> </p>
<p><span m="811360">Now</span> <span m="812150">in</span> <span m="812270">1637,</span> <span m="814440">Fermat</span> <span m="815060">wrote</span> <span m="815290">in</span> <span m="815340">the</span> <span m="815420">margin</span> <span m="815830">of</span> <span m="815910">a</span> <span m="815980">block</span> <span m="817030">that</span> <span m="817170">he had</span> <span m="817420">discovered</span> <span m="817950">a</span> <span m="818010">proof</span> <span m="818360">of</span> <span m="818440">this</span> <span m="818620">result,</span> <span m="819410">but</span> <span m="819590">the</span> <span m="819680">proof</span> <span m="820070">was</span> <span m="820210">too</span> <span m="820390">long</span> <span m="820760">to</span> <span m="820830">fit</span> <span m="821140">in</span> <span m="821250">the</span> <span m="821320">margin.</span> <span m="822510">And</span> <span m="822610">so</span> <span m="822720">he</span> <span m="822830">wasn't</span> <span m="823200">going</span> <span m="823310">to</span> <span m="823380">supply</span> <span m="823780">the</span> <span m="823890">details</span> <span m="824400">there.</span> <span m="825870">And</span> <span m="826130">he</span> <span m="826220">never</span> <span m="826470">did</span> <span m="826710">supply</span> <span m="827050">the</span> <span m="827150">details.</span> </p>
<p><span m="827730">And</span> <span m="827820">it</span> <span m="827940">took</span> <span m="828160">350</span> <span m="829050">years</span> <span m="830240">and</span> <span m="830430">100s</span> <span m="831060">and</span> <span m="831150">100s</span> <span m="831510">of</span> <span m="831610">pages.</span> <span m="832750">A</span> <span m="832820">fellow</span> <span m="833150">named</span> <span m="833480">Andrew</span> <span m="833910">Wiles--</span> <span m="834910">took</span> <span m="835100">him</span> <span m="835320">personally</span> <span m="835860">over</span> <span m="836050">10</span> <span m="836330">years.</span> <span m="836710">He</span> <span m="837010">more or</span> <span m="837260">less</span> <span m="837480">locked</span> <span m="837800">himself</span> <span m="838180">in</span> <span m="838250">his</span> <span m="838360">room</span> <span m="838620">for</span> <span m="838740">10</span> <span m="839000">years,</span> <span m="840240">produced</span> <span m="840620">100s</span> <span m="840870">of</span> <span m="840960">pages,</span> <span m="841360">and</span> <span m="841460">he</span> <span m="841550">finally</span> <span m="841900">did</span> <span m="842140">prove</span> <span m="842420">Fermat's</span> <span m="842810">Last</span> <span m="843030">Theorem.</span> </p>
<p><span m="844350">So</span> <span m="844830">Fermat</span> <span m="845200">was</span> <span m="845320">right</span> <span m="845480">about</span> <span m="845670">one</span> <span m="845900">thing--</span> <span m="846520">it</span> <span m="846660">would</span> <span m="846820">not</span> <span m="847200">fit</span> <span m="847450">in</span> <span m="847570">the</span> <span m="847640">margin.</span> <span m="848280">At</span> <span m="848370">least,</span> <span m="848790">there's</span> <span m="848970">no</span> <span m="849120">proof</span> <span m="849400">that</span> <span m="849470">we</span> <span m="849640">know of</span> <span m="850030">that would</span> <span m="850160">fit in</span> <span m="850370">the</span> <span m="850450">margin of a</span> <span m="850920">book.</span> <span m="852990">Any</span> <span m="853200">questions</span> <span m="853720">on</span> <span m="854070">proof</span> <span m="854420">technique?</span> <span m="856480">We're</span> <span m="856650">going</span> <span m="856760">to</span> <span m="856850">pound</span> <span m="857430">a</span> <span m="857480">lot</span> <span m="857760">on</span> <span m="857920">this</span> <span m="859720">over</span> <span m="859880">the</span> <span m="859970">next</span> <span m="860220">few</span> <span m="860380">weeks.</span> </p>
<p><span m="864060">All</span> <span m="864130">right,</span> <span m="864360">next</span> <span m="864640">we're</span> <span m="864710">going</span> <span m="864830">to</span> <span m="865430">look</span> <span m="865650">at</span> <span m="865720">a</span> <span m="865770">class</span> <span m="866150">of</span> <span m="866270">puzzles</span> <span m="868050">that</span> <span m="868180">was</span> <span m="868330">very</span> <span m="868700">popular</span> <span m="869550">in</span> <span m="869650">the</span> <span m="869760">late</span> <span m="870110">1800s.</span> <span m="871850">Now,</span> <span m="872090">in</span> <span m="872210">these</span> <span m="872480">puzzles,</span> <span m="873090">you</span> <span m="873200">have</span> <span m="873430">a</span> <span m="873520">grid</span> <span m="874450">of</span> <span m="874560">letters</span> <span m="875130">or</span> <span m="875300">numbers.</span> <span m="876390">And</span> <span m="876500">you've</span> <span m="876580">got</span> <span m="876720">to</span> <span m="876780">slide</span> <span m="877420">the</span> <span m="877500">letters</span> <span m="877870">and</span> <span m="877950">numbers</span> <span m="878280">around</span> <span m="879130">to</span> <span m="879490">put</span> <span m="879740">them</span> <span m="880070">in</span> <span m="880220">order.</span> </p>
<p><span m="881750">And</span> <span m="881900">so</span> <span m="882050">as</span> <span m="882160">an</span> <span m="882220">example,</span> <span m="885190">here's</span> <span m="885500">the</span> <span m="885590">problem</span> <span m="886050">on</span> <span m="886160">a</span> <span m="886220">three</span> <span m="886440">by</span> <span m="886620">three</span> <span m="886970">grid,</span> <span m="887520">or an</span> <span m="887620">eight</span> <span m="887790">puzzle.</span> <span m="890050">You</span> <span m="890170">want</span> <span m="890310">to</span> <span m="890380">find</span> <span m="892030">a</span> <span m="892260">sequence</span> <span m="892790">of</span> <span m="892910">moves</span> <span m="900570">to</span> <span m="900700">go</span> <span m="903740">from</span> <span m="905510">this</span> <span m="905760">configuration.</span> <span m="907160">I'll</span> <span m="907260">do</span> <span m="907410">a</span> <span m="907470">three</span> <span m="907670">by</span> <span m="907830">three</span> <span m="908100">case.</span> </p>
<p><span m="910370">I</span> <span m="910460">have</span> <span m="910690">a</span> <span m="911000">here,</span> <span m="911500">b</span> <span m="911850">here,</span> <span m="912230">c,</span> <span m="913096">d,</span> <span m="913529">e,</span> <span m="914395">f.</span> <span m="915060">And</span> <span m="915220">I've</span> <span m="915290">put</span> <span m="916340">g</span> <span m="916610">and</span> <span m="916700">h</span> <span m="916970">out</span> <span m="917080">of</span> <span m="917170">order.</span> <span m="918530">And</span> <span m="918680">that</span> <span m="918870">square is</span> <span m="919250">blank,</span> <span m="920050">and</span> <span m="920160">I</span> <span m="920230">can</span> <span m="920410">move</span> <span m="921920">tiles</span> <span m="922440">or</span> <span m="922480">letters</span> <span m="922820">into</span> <span m="923120">it.</span> </p>
<p><span m="923600">I'm going</span> <span m="923790">to</span> <span m="923810">start</span> <span m="924170">here.</span> <span m="924670">And</span> <span m="924830">I</span> <span m="924910">eventually</span> <span m="925360">want</span> <span m="925530">to</span> <span m="925570">get</span> <span m="925870">to</span> <span m="928240">the</span> <span m="928360">configuration</span> <span m="929170">where</span> <span m="929260">the</span> <span m="929320">letters</span> <span m="929710">are</span> <span m="929850">all</span> <span m="930590">in</span> <span m="930730">alphabetical</span> <span m="931340">order.</span> <span m="935720">So</span> <span m="935920">g and</span> <span m="936300">h</span> <span m="936565">are</span> <span m="936830">in</span> <span m="936990">order.</span> <span m="940290">Now,</span> <span m="941120">a legal</span> <span m="941600">move</span> <span m="942020">means</span> <span m="942230">you</span> <span m="942330">slide</span> <span m="942910">a</span> <span m="942960">letter</span> <span m="943430">into</span> <span m="943700">an</span> <span m="943770">adjacent</span> <span m="944280">blank</span> <span m="944550">square.</span> </p>
<p></p>
<p><span m="958370">And</span> <span m="958550">you</span> <span m="958640">can</span> <span m="958770">go</span> <span m="958950">in</span> <span m="959020">a</span> <span m="959140">row</span> <span m="959510">or</span> <span m="959710">a</span> <span m="959810">column.</span> </p>
<p></p>
<p><span m="970240">OK,</span> <span m="970500">everybody</span> <span m="970670">understand</span> <span m="971090">the</span> <span m="971170">game,</span> <span m="971870">what</span> <span m="971990">you're</span> <span m="972110">supposed</span> <span m="972390">to</span> <span m="972470">do?</span> <span m="973680">You</span> <span m="973790">slide</span> <span m="974220">these</span> <span m="974470">things.</span> <span m="974810">That</span> <span m="975010">could</span> <span m="975130">go</span> <span m="975270">down.</span> <span m="975680">That</span> <span m="975830">could</span> <span m="975920">go</span> <span m="976080">across.</span> </p>
<p><span m="976600">How</span> <span m="976680">many</span> <span m="976830">people</span> <span m="977120">played</span> <span m="977430">a</span> <span m="977500">game</span> <span m="977770">like</span> <span m="977980">this</span> <span m="978160">before?</span> <span m="979450">Good,</span> <span m="980350">a</span> <span m="980400">lot</span> <span m="980690">of</span> <span m="980770">you.</span> <span m="981040">OK,</span> <span m="981790">so</span> <span m="981960">I</span> <span m="982070">need</span> <span m="982280">some</span> <span m="982420">volunteers</span> <span m="983160">who</span> <span m="983270">are</span> <span m="983360">good</span> <span m="983710">at</span> <span m="983900">solving</span> <span m="984350">puzzles</span> <span m="985950">that</span> <span m="986070">can</span> <span m="986250">do</span> <span m="986440">this,</span> <span m="987120">have</span> <span m="987590">played</span> <span m="987950">games</span> <span m="988320">like</span> <span m="988590">this,</span> <span m="989140">that</span> <span m="989280">are</span> <span m="989340">pretty</span> <span m="989610">quick</span> <span m="989900">at</span> <span m="990050">it.</span> </p>
<p><span m="994490">So</span> <span m="994600">I've</span> <span m="994710">got</span> <span m="994930">it</span> <span m="995010">here.</span> <span m="997930">Who</span> <span m="998320">would</span> <span m="998440">like</span> <span m="998600">to</span> <span m="998690">come</span> <span m="998890">down?</span> <span m="999240">There are</span> <span m="999570">prizes</span> <span m="1000080">if</span> <span m="1000170">you</span> <span m="1000270">can</span> <span m="1000410">solve</span> <span m="1000670">this.</span> </p>
<p><span m="1000790">What we're</span> <span m="1000970">going</span> <span m="1001060">to</span> <span m="1001140">do</span> <span m="1002020">is</span> <span m="1002350">solve</span> <span m="1002650">this.</span> <span m="1002820">And</span> <span m="1003030">see</span> <span m="1003120">if</span> <span m="1003190">you</span> <span m="1003240">can</span> <span m="1003330">solve</span> <span m="1003630">it</span> <span m="1003670">in</span> <span m="1003740">three</span> <span m="1004040">minutes.</span> <span m="1004530">But</span> <span m="1004650">I</span> <span m="1004710">need</span> <span m="1005440">three</span> <span m="1005650">students</span> <span m="1006050">to</span> <span m="1006140">come</span> <span m="1006320">on</span> <span m="1006470">down</span> <span m="1006680">and</span> <span m="1006770">do</span> <span m="1006910">this.</span> </p>
<p><span m="1008170">Who'd</span> <span m="1008340">like</span> <span m="1008520">to</span> <span m="1008610">volunteer?</span> <span m="1009360">Who</span> <span m="1009480">thinks</span> <span m="1009700">they</span> <span m="1009900">can</span> <span m="1010190">handle</span> <span m="1010510">puzzles</span> <span m="1011470">under</span> <span m="1011660">pressure?</span> <span m="1012665">So you</span> <span m="1013260">have</span> <span m="1013360">the</span> <span m="1013450">g and</span> <span m="1013740">h</span> <span m="1014030">out of</span> <span m="1014110">order.</span> <span m="1014470">So</span> <span m="1014700">it's</span> <span m="1014830">not</span> <span m="1014950">so</span> <span m="1015080">hard.</span> </p>
<p><span m="1015695">You</span> <span m="1016010">move</span> <span m="1016270">the f</span> <span m="1016420">down</span> <span m="1017190">there.</span> <span m="1018960">I</span> <span m="1019080">don't</span> <span m="1019220">know,</span> <span m="1020430">play</span> <span m="1020580">around</span> <span m="1020830">like</span> <span m="1020990">that.</span> <span m="1023070">So</span> <span m="1023190">I</span> <span m="1023260">need</span> <span m="1023430">some</span> <span m="1023540">volunteers.</span> <span m="1024060">Who'd</span> <span m="1024150">like to</span> <span m="1024369">come</span> <span m="1024560">down</span> <span m="1024790">and</span> <span m="1024869">do</span> <span m="1024990">this?</span> </p>
<p><span m="1027210">You</span> <span m="1027339">can</span> <span m="1027460">win</span> <span m="1027670">candy.</span> <span m="1028089">I've</span> <span m="1028290">got</span> <span m="1028589">candy</span> <span m="1028890">here.</span> <span m="1031589">There's</span> <span m="1031760">even</span> <span m="1031940">a</span> <span m="1032000">prize</span> <span m="1032410">if</span> <span m="1032520">you</span> <span m="1032619">don't</span> <span m="1032819">make</span> <span m="1033030">it</span> <span m="1033119">in</span> <span m="1033230">three</span> <span m="1033450">minutes.</span> <span m="1034609">Who</span> <span m="1034710">wants</span> <span m="1034920">some</span> <span m="1035079">candy?</span> <span m="1037109">Nobody?</span> </p>
<p><span m="1038076">Is there a prize if we</span> <span m="1038553">don't make it?</span> </p>
<p><span m="1039030">What's</span> <span m="1039230">that?</span> </p>
<p><span m="1039940">Is there a prize if</span> <span m="1040350">we don't make it?</span> </p>
<p><span m="1041170">There</span> <span m="1041390">is.</span> <span m="1041650">Come</span> <span m="1041819">on</span> <span m="1041970">down.</span> <span m="1042750">All right,</span> <span m="1043569">now,</span> <span m="1043700">who's</span> <span m="1043900">going</span> <span m="1044010">to</span> <span m="1044069">help</span> <span m="1044270">him</span> <span m="1044430">out?</span> <span m="1047250">I need</span> <span m="1047440">a</span> <span m="1047500">couple</span> <span m="1047730">more</span> <span m="1047920">people</span> <span m="1048220">here</span> <span m="1048369">just</span> <span m="1048650">to</span> <span m="1048790">mess him</span> <span m="1049130">up,</span> <span m="1049490">because</span> <span m="1049890">by</span> <span m="1050060">himself--</span> <span m="1050360">there</span> <span m="1050430">we</span> <span m="1050750">go--</span> <span m="1050960">good.</span> <span m="1051520">Anybody</span> <span m="1051830">else</span> <span m="1052050">like</span> <span m="1052230">to</span> <span m="1052380">come</span> <span m="1052520">on</span> <span m="1052650">down?</span> <span m="1052850">Because if</span> <span m="1052960">we</span> <span m="1053070">get</span> <span m="1053260">three,</span> <span m="1053580">it</span> <span m="1053710">really</span> <span m="1053910">gets</span> <span m="1054070">hard,</span> <span m="1054780">because</span> <span m="1054960">you</span> <span m="1055040">fight</span> <span m="1055300">over</span> <span m="1055460">what</span> <span m="1055640">moves</span> <span m="1055880">to</span> <span m="1055970">make.</span> </p>
<p><span m="1057070">Now,</span> <span m="1057780">what</span> <span m="1057980">I'd</span> <span m="1058060">like</span> <span m="1058250">you</span> <span m="1058390">all</span> <span m="1058560">to</span> <span m="1058640">do</span> <span m="1058960">is</span> <span m="1059090">sort</span> <span m="1059310">of</span> <span m="1059370">scream</span> <span m="1059800">out</span> <span m="1060040">possibilities</span> <span m="1060750">for</span> <span m="1061000">him.</span> <span m="1061290">Like he figures it</span> <span m="1061550">out</span> <span m="1061670">there,</span> <span m="1061810">you</span> <span m="1061960">scream</span> <span m="1062290">out,</span> <span m="1062450">that'll</span> <span m="1062680">really</span> <span m="1063510">screw</span> <span m="1063720">him</span> <span m="1063850">up.</span> <span m="1064000">OK,</span> <span m="1065480">now,</span> <span m="1065570">you</span> <span m="1065670">can't</span> <span m="1065920">pick</span> <span m="1066050">them</span> <span m="1066100">up.</span> <span m="1066420">You've just</span> <span m="1066540">got</span> <span m="1066650">to</span> <span m="1066690">slide</span> <span m="1066940">them</span> <span m="1067040">around.</span> </p>
<p><span m="1067880">And</span> <span m="1068050">you</span> <span m="1068150">can</span> <span m="1068290">go,</span> <span m="1068550">and</span> <span m="1068620">we'll</span> <span m="1068720">go</span> <span m="1068840">to</span> <span m="1068930">2:56,</span> <span m="1070510">and</span> <span m="1070700">see</span> <span m="1070830">if</span> <span m="1070890">you</span> <span m="1070990">can</span> <span m="1071120">solve</span> <span m="1071430">it.</span> <span m="1071490">If</span> <span m="1071560">you</span> <span m="1071680">can solve</span> <span m="1071970">it,</span> <span m="1072130">you</span> <span m="1072210">guys</span> <span m="1072460">get</span> <span m="1074000">candy.</span> <span m="1077070">And</span> <span m="1077430">if</span> <span m="1077560">you</span> <span m="1077660">can</span> <span m="1077780">figure</span> <span m="1078020">it</span> <span m="1078220">out out</span> <span m="1078380">there,</span> <span m="1078660">let</span> <span m="1078780">them</span> <span m="1078880">know.</span> </p>
<p><span m="1079723">Slide to</span> <span m="1080216">the gray area.</span> </p>
<p><span m="1082190">No,</span> <span m="1082360">you</span> <span m="1082490">can't</span> <span m="1082730">use</span> <span m="1082920">the</span> <span m="1083010">gray</span> <span m="1083290">area.</span> <span m="1087200">All right,</span> <span m="1087470">you've got</span> <span m="1087590">about</span> <span m="1087800">two</span> <span m="1087920">minutes.</span> <span m="1088270">They</span> <span m="1088360">have</span> <span m="1088470">the</span> <span m="1088550">g and h</span> <span m="1088810">in</span> <span m="1089160">order.</span> <span m="1090910">Everything</span> <span m="1091170">else</span> <span m="1091350">is</span> <span m="1091440">pretty</span> <span m="1091620">well</span> <span m="1091770">screwed</span> <span m="1092010">up,</span> <span m="1092160">though.</span> </p>
<p><span m="1095360">What</span> <span m="1095510">do you</span> <span m="1095580">got?</span> <span m="1095760">A, b,</span> <span m="1095930">c, d,</span> <span m="1096406">e,</span> <span m="1096882">f,</span> <span m="1097360">h</span> <span m="1097645">g--</span> <span m="1097930">no,</span> <span m="1098040">no,</span> <span m="1098140">no,</span> <span m="1098290">that's</span> <span m="1098490">where</span> <span m="1098600">you</span> <span m="1098710">started,</span> <span m="1099100">guys.</span> <span m="1099400">That's</span> <span m="1100200">not</span> <span m="1100430">it.</span> <span m="1101750">It was</span> <span m="1101880">close,</span> <span m="1101980">just</span> <span m="1102740">one</span> <span m="1102930">was</span> <span m="1103070">out</span> <span m="1103190">of</span> <span m="1103270">order</span> <span m="1103510">is</span> <span m="1103620">all.</span> <span m="1108150">Just</span> <span m="1108400">one</span> <span m="1108650">letter</span> <span m="1108950">at</span> <span m="1109010">a</span> <span m="1109080">time.</span> <span m="1109850">There</span> <span m="1110110">you</span> <span m="1110150">go.</span> </p>
<p></p>
<p><span m="1123160">I</span> <span m="1123260">think</span> <span m="1123420">they</span> <span m="1123520">need</span> <span m="1123730">help.</span> <span m="1124060">We</span> <span m="1124140">got</span> <span m="1124290">about</span> <span m="1124480">a</span> <span m="1124520">minute</span> <span m="1124830">ago.</span> </p>
<p><span m="1125710">Cannot see anything.</span> </p>
<p><span m="1126950">You</span> <span m="1127070">can't</span> <span m="1127340">see</span> <span m="1127510">anything.</span> <span m="1127850">Well,</span> <span m="1128330">I</span> <span m="1128490">don't</span> <span m="1128590">know</span> <span m="1128700">if</span> <span m="1128780">it</span> <span m="1128840">would</span> <span m="1128960">help.</span> <span m="1132310">You</span> <span m="1132410">want</span> <span m="1132530">to</span> <span m="1132580">show</span> <span m="1132820">them where</span> <span m="1132990">you are?</span> </p>
<p></p>
<p><span m="1139510">This</span> <span m="1139680">is</span> <span m="1139780">not</span> <span m="1140000">looking</span> <span m="1140290">good.</span> </p>
<p></p>
<p><span m="1151260">We're</span> <span m="1151430">pretty</span> <span m="1151630">close</span> <span m="1151890">to</span> <span m="1151960">the</span> <span m="1152090">end</span> <span m="1152240">here,</span> <span m="1152390">guys.</span> <span m="1157610">A,</span> <span m="1157750">b,</span> <span m="1158205">c,</span> <span m="1159115">f, e,</span> <span m="1159570">d,</span> <span m="1160394">g, h--</span> <span m="1160806">g,</span> <span m="1161220">h</span> <span m="1161410">is</span> <span m="1161510">good.</span> <span m="1163020">F</span> <span m="1163370">is</span> <span m="1163500">not</span> <span m="1163650">so</span> <span m="1163770">good.</span> <span m="1167140">All right,</span> <span m="1167540">you got</span> <span m="1167680">about</span> <span m="1168190">30</span> <span m="1168480">seconds.</span> </p>
<p></p>
<p><span m="1185030">Whoa,</span> <span m="1185310">what</span> <span m="1185590">are</span> <span m="1185780">you</span> <span m="1185940">guys</span> <span m="1185980">doing?</span> <span m="1186530">One</span> <span m="1186930">at a</span> <span m="1187030">time</span> <span m="1187370">here.</span> <span m="1187740">One</span> <span m="1188120">at</span> <span m="1188240">the</span> <span m="1188350">time.</span> <span m="1189566">In</span> <span m="1189950">fact,</span> <span m="1190370">I</span> <span m="1190490">think</span> <span m="1192470">time</span> <span m="1192680">is</span> <span m="1192880">up</span> <span m="1193090">here.</span> </p>
<p></p>
<p><span m="1202940">All</span> <span m="1203280">right,</span> <span m="1204160">2:56.</span> <span m="1204640">Oh,</span> <span m="1205420">no,</span> <span m="1205600">no.</span> <span m="1207020">That</span> <span m="1207470">we</span> <span m="1207600">don't</span> <span m="1207850">do.</span> <span m="1208110">No</span> <span m="1208880">diagonal</span> <span m="1209440">moves--</span> <span m="1210310">no,</span> <span m="1210600">no,</span> <span m="1210710">no,</span> <span m="1210920">no.</span> </p>
<p><span m="1212020">All</span> <span m="1212390">right,</span> <span m="1212850">well,</span> <span m="1213570">you</span> <span m="1213740">guys</span> <span m="1213990">were</span> <span m="1214050">good</span> <span m="1214360">sports,</span> <span m="1214680">so</span> <span m="1215320">you</span> <span m="1215470">do</span> <span m="1215600">get</span> <span m="1215750">a</span> <span m="1215810">prize.</span> <span m="1216950">It's</span> <span m="1217050">not</span> <span m="1217230">the</span> <span m="1217320">candy,</span> <span m="1218150">but</span> <span m="1218300">I</span> <span m="1218420">have</span> <span m="1218620">these</span> <span m="1218800">wonderful</span> <span m="1219440">plastic</span> <span m="1219930">Nerd</span> <span m="1220200">Pride</span> <span m="1220600">pocket</span> <span m="1220930">protectors</span> <span m="1222060">for</span> <span m="1222310">you.</span> <span m="1223230">There you</span> <span m="1223580">go, very</span> <span m="1223910">good</span> <span m="1224050">sports.</span> <span m="1224430">Thank</span> <span m="1224710">you,</span> <span m="1225480">well</span> <span m="1225730">done.</span> <span m="1227570">All right.</span> </p>
<p><span m="1228400">[APPLAUSE]</span> </p>
<p><span m="1232700">Now,</span> <span m="1233390">I</span> <span m="1233540">was</span> <span m="1233870">a</span> <span m="1233940">little</span> <span m="1234180">cruel</span> <span m="1234510">to</span> <span m="1234610">these</span> <span m="1234840">guys,</span> <span m="1235780">for</span> <span m="1236040">a</span> <span m="1236090">lot</span> <span m="1236250">of</span> <span m="1236310">reasons.</span> <span m="1237780">It's</span> <span m="1237990">impossible?</span> <span m="1239830">You</span> <span m="1239940">think?</span> </p>
<p><span m="1241162">[INDISTINCT CHATTER]</span> </p>
<p><span m="1242930">Yeah</span> <span m="1243330">it</span> <span m="1243420">is</span> <span m="1243640">impossible.</span> <span m="1245590">This</span> <span m="1245760">is</span> <span m="1245850">impossible</span> <span m="1246360">to</span> <span m="1246470">do.</span> <span m="1247610">Let's</span> <span m="1247870">see</span> <span m="1247980">if</span> <span m="1248090">we</span> <span m="1248190">can</span> <span m="1248350">prove</span> <span m="1248770">it's</span> <span m="1248920">impossible.</span> <span m="1250470">Because</span> <span m="1251090">it</span> <span m="1251160">doesn't</span> <span m="1251470">seem</span> <span m="1252260">so</span> <span m="1253010">hard,</span> <span m="1253280">necessarily.</span> </p>
<p><span m="1255230">But</span> <span m="1255410">let's</span> <span m="1256010">take</span> <span m="1256170">that</span> <span m="1256320">as</span> <span m="1256420">a</span> <span m="1256490">theorem.</span> <span m="1262320">There</span> <span m="1262900">is</span> <span m="1263430">no</span> <span m="1266170">sequence</span> <span m="1272510">of</span> <span m="1272630">legal</span> <span m="1275380">moves--</span> <span m="1276990">diagonal</span> <span m="1277490">moves</span> <span m="1277770">makes</span> <span m="1278020">this</span> <span m="1278130">a</span> <span m="1278190">lot</span> <span m="1278390">easier--</span> <span m="1280650">to</span> <span m="1281050">invert</span> <span m="1283780">g</span> <span m="1284350">and</span> <span m="1284540">h</span> <span m="1286680">and</span> <span m="1288000">also</span> <span m="1288360">return</span> <span m="1290990">all</span> <span m="1291270">the</span> <span m="1291420">other</span> <span m="1291690">letters</span> <span m="1295820">to</span> <span m="1295920">their</span> <span m="1296120">original</span> <span m="1296710">order</span> <span m="1297530">or</span> <span m="1297660">position.</span> </p>
<p></p>
<p><span m="1309400">Now,</span> <span m="1309790">to</span> <span m="1309880">prove</span> <span m="1310230">this,</span> <span m="1311620">we're</span> <span m="1311850">going</span> <span m="1312120">to</span> <span m="1312220">use</span> <span m="1313790">what's</span> <span m="1313990">called</span> <span m="1314290">an</span> <span m="1314390">invariant.</span> <span m="1316530">It's</span> <span m="1316710">a</span> <span m="1316760">very</span> <span m="1317370">powerful</span> <span m="1318040">and</span> <span m="1318120">commonly</span> <span m="1318620">used</span> <span m="1318910">concept</span> <span m="1319420">in</span> <span m="1319480">computer</span> <span m="1319870">science,</span> <span m="1320820">very</span> <span m="1321100">closely</span> <span m="1321590">tied</span> <span m="1321900">to</span> <span m="1322020">induction,</span> <span m="1322470">as</span> <span m="1322580">we'll</span> <span m="1322710">see.</span> <span m="1324500">In</span> <span m="1324630">order</span> <span m="1324880">to</span> <span m="1324950">show</span> <span m="1325480">that</span> <span m="1325630">your</span> <span m="1325810">system</span> <span m="1327270">can</span> <span m="1327480">never</span> <span m="1327800">reach</span> <span m="1328320">a</span> <span m="1328410">particular</span> <span m="1329070">special</span> <span m="1329540">state,</span> <span m="1331580">it</span> <span m="1331710">is</span> <span m="1331810">sufficient</span> <span m="1332280">to</span> <span m="1332340">show</span> <span m="1332570">there's</span> <span m="1332770">some</span> <span m="1333000">property</span> <span m="1333600">called</span> <span m="1333900">the</span> <span m="1333990">invariant</span> <span m="1335160">that</span> <span m="1335310">holds</span> <span m="1335980">at</span> <span m="1336170">the</span> <span m="1336280">initial</span> <span m="1336630">state,</span> <span m="1337880">and</span> <span m="1338030">that</span> <span m="1338140">is</span> <span m="1338270">preserved</span> <span m="1338990">by</span> <span m="1339130">every</span> <span m="1339480">legal</span> <span m="1339820">move,</span> <span m="1341540">and</span> <span m="1342720">is</span> <span m="1342910">not</span> <span m="1343210">present--</span> <span m="1343800">does</span> <span m="1343960">not</span> <span m="1344180">hold--</span> <span m="1344760">in</span> <span m="1344890">that</span> <span m="1345100">special</span> <span m="1345510">state.</span> <span m="1346770">The</span> <span m="1346910">idea</span> <span m="1347320">is,</span> <span m="1347630">if you</span> <span m="1347750">get</span> <span m="1347930">this</span> <span m="1348090">magic</span> <span m="1348550">property</span> <span m="1348980">called</span> <span m="1349230">the</span> <span m="1349320">invariant</span> <span m="1349780">and</span> <span m="1350020">it</span> <span m="1350150">holds</span> <span m="1350450">at the</span> <span m="1350620">start,</span> <span m="1351540">and</span> <span m="1351720">it</span> <span m="1351800">holds</span> <span m="1352200">across</span> <span m="1352590">every</span> <span m="1352890">step,</span> <span m="1354020">then</span> <span m="1354150">the</span> <span m="1354330">only</span> <span m="1354730">states</span> <span m="1355070">you</span> <span m="1355200">can</span> <span m="1355340">reach</span> <span m="1356450">have</span> <span m="1356640">to</span> <span m="1356720">have</span> <span m="1357590">to</span> <span m="1357660">have</span> <span m="1357790">the</span> <span m="1357870">property,</span> <span m="1358210">be</span> <span m="1358320">invariant.</span> <span m="1359450">And</span> <span m="1359570">if</span> <span m="1359700">the</span> <span m="1359760">special</span> <span m="1360240">state</span> <span m="1360940">doesn't</span> <span m="1361350">have</span> <span m="1361960">that</span> <span m="1362150">property,</span> <span m="1362850">you</span> <span m="1362980">can't</span> <span m="1363280">reach</span> <span m="1363540">it.</span> </p>
<p><span m="1365060">All</span> <span m="1365140">right,</span> <span m="1365840">so</span> <span m="1367360">there</span> <span m="1367640">is</span> <span m="1367750">going</span> <span m="1367850">to</span> <span m="1367890">be</span> <span m="1367970">some</span> <span m="1368280">property</span> <span m="1368750">we're</span> <span m="1368880">going</span> <span m="1368960">to</span> <span m="1369030">look</span> <span m="1369280">for</span> <span m="1369550">that</span> <span m="1369790">held</span> <span m="1370130">at</span> <span m="1370220">the</span> <span m="1370300">beginning</span> <span m="1370740">here--</span> <span m="1371680">and</span> <span m="1371750">the</span> <span m="1371820">beginning</span> <span m="1372290">had</span> <span m="1372740">this</span> <span m="1372950">set</span> <span m="1373110">up--</span> <span m="1376440">that</span> <span m="1376620">holds,</span> <span m="1377700">is</span> <span m="1377900">preserved,</span> <span m="1378520">by</span> <span m="1378650">every</span> <span m="1378970">move,</span> <span m="1379420">every</span> <span m="1379630">legal</span> <span m="1379880">move,</span> <span m="1380630">but</span> <span m="1380840">does</span> <span m="1381030">not</span> <span m="1381350">hold</span> <span m="1382320">in</span> <span m="1382500">this</span> <span m="1382750">state.</span> <span m="1383800">And</span> <span m="1383990">therefore,</span> <span m="1384380">you</span> <span m="1384510">can</span> <span m="1384650">never</span> <span m="1385030">reach</span> <span m="1385330">this</span> <span m="1385540">state</span> <span m="1385840">legally.</span> <span m="1386770">So</span> <span m="1386810">they</span> <span m="1386920">were</span> <span m="1387050">doomed.</span> </p>
<p><span m="1389880">So</span> <span m="1390010">we've</span> <span m="1390100">got</span> <span m="1390270">to</span> <span m="1390330">figure</span> <span m="1390670">out</span> <span m="1391350">what</span> <span m="1391570">that</span> <span m="1391760">property</span> <span m="1392280">is.</span> <span m="1395240">And</span> <span m="1395330">that's</span> <span m="1395520">always</span> <span m="1395820">the</span> <span m="1395920">trick</span> <span m="1396710">in</span> <span m="1397130">analyzing</span> <span m="1397730">systems</span> <span m="1398340">or</span> <span m="1398470">algorithms--</span> <span m="1399100">is</span> <span m="1399400">what's</span> <span m="1399790">that</span> <span m="1400020">key</span> <span m="1400200">property,</span> <span m="1400830">the</span> <span m="1401020">invariant</span> <span m="1401250">property?</span> <span m="1402960">So</span> <span m="1403090">to</span> <span m="1403180">figure</span> <span m="1403560">out</span> <span m="1403770">the</span> <span m="1403870">invariant,</span> <span m="1404510">we sort of</span> <span m="1404680">have</span> <span m="1404800">to</span> <span m="1404900">look</span> <span m="1405200">at</span> <span m="1406030">what</span> <span m="1406280">happens</span> <span m="1406690">during</span> <span m="1406970">a</span> <span m="1407020">move</span> <span m="1407700">in</span> <span m="1407830">this</span> <span m="1408040">system--</span> <span m="1408400">a</span> <span m="1408480">transition.</span> </p>
<p><span m="1410080">Well,</span> <span m="1410560">there's</span> <span m="1410700">two</span> <span m="1410870">kinds</span> <span m="1411190">of</span> <span m="1411270">moves.</span> <span m="1413130">There's</span> <span m="1413340">a</span> <span m="1413440">row</span> <span m="1413710">move.</span> <span m="1417860">And an</span> <span m="1418020">example</span> <span m="1418490">of</span> <span m="1418540">a</span> <span m="1418620">row</span> <span m="1418850">move</span> <span m="1425050">is</span> <span m="1425320">you</span> <span m="1425400">might</span> <span m="1425590">have</span> <span m="1425810">a,</span> <span m="1426160">b, c,</span> <span m="1429100">d,</span> <span m="1430080">g,</span> <span m="1431360">e, f,</span> <span m="1431750">h.</span> </p>
<p><span m="1433560">And</span> <span m="1433730">I'm</span> <span m="1433850">going</span> <span m="1433960">to</span> <span m="1434000">move</span> <span m="1434270">the g</span> <span m="1435370">its</span> <span m="1435520">row</span> <span m="1435830">to the</span> <span m="1435950">blank</span> <span m="1436250">square.</span> <span m="1437810">So</span> <span m="1438050">it</span> <span m="1438160">would</span> <span m="1438290">become</span> <span m="1439060">this</span> <span m="1439340">state.</span> <span m="1445010">So</span> <span m="1445180">g</span> <span m="1445550">moves</span> <span m="1445830">rightward</span> <span m="1446360">into</span> <span m="1446410">the</span> <span m="1446510">blank,</span> <span m="1447720">and</span> <span m="1447840">the</span> <span m="1447910">rest</span> <span m="1448230">stays</span> <span m="1448450">the</span> <span m="1448510">same.</span> </p>
<p><span m="1451020">Now,</span> <span m="1451250">when</span> <span m="1451400">I</span> <span m="1451470">make</span> <span m="1451720">a</span> <span m="1451820">row</span> <span m="1452070">move,</span> <span m="1454250">did</span> <span m="1454510">the</span> <span m="1454750">relative</span> <span m="1455430">order</span> <span m="1456150">of</span> <span m="1456270">the</span> <span m="1456430">items</span> <span m="1457770">change?</span> <span m="1461060">No.</span> <span m="1461870">Of</span> <span m="1461940">course,</span> <span m="1462160">to be</span> <span m="1462280">precise,</span> <span m="1462680">I</span> <span m="1462720">better</span> <span m="1463040">define</span> <span m="1463630">what</span> <span m="1463810">relative</span> <span m="1465010">order</span> <span m="1465340">means,</span> <span m="1465700">or</span> <span m="1465760">the</span> <span m="1465870">natural</span> <span m="1466260">order.</span> <span m="1467280">So</span> <span m="1467970">by</span> <span m="1468260">the</span> <span m="1468330">natural</span> <span m="1468770">order,</span> <span m="1472920">I</span> <span m="1473050">mean</span> <span m="1473660">this</span> <span m="1473940">ordering--</span> <span m="1477350">1, 2</span> <span m="1477815">3,</span> <span m="1478280">4,</span> <span m="1478745">5,</span> <span m="1479210">6,</span> <span m="1480140">7,</span> <span m="1480605">8,</span> <span m="1481070">9.</span> </p>
<p><span m="1482100">So</span> <span m="1482210">if</span> <span m="1482330">I</span> <span m="1482450">look</span> <span m="1482820">at</span> <span m="1483410">the</span> <span m="1483560">order</span> <span m="1483950">here,</span> <span m="1485080">g</span> <span m="1485370">moved</span> <span m="1485750">from</span> <span m="1485940">5</span> <span m="1486360">to</span> <span m="1486450">6,</span> <span m="1487070">but</span> <span m="1487590">didn't</span> <span m="1487840">change</span> <span m="1488300">order</span> <span m="1488610">with</span> <span m="1488760">respect</span> <span m="1489090">to</span> <span m="1489150">any</span> <span m="1489440">of</span> <span m="1489480">the</span> <span m="1489570">other</span> <span m="1489720">guys.</span> <span m="1492900">Let's</span> <span m="1493100">state</span> <span m="1493330">that</span> <span m="1493520">as</span> <span m="1493640">a</span> <span m="1493710">lemma.</span> <span m="1502780">Because</span> <span m="1502930">that'll</span> <span m="1503940">be</span> <span m="1504040">very</span> <span m="1504250">useful.</span> </p>
<p><span m="1505490">Lemma</span> <span m="1505690">1--</span> <span m="1507740">a</span> <span m="1507850">row</span> <span m="1508150">move</span> <span m="1512060">does</span> <span m="1512320">not</span> <span m="1512680">change</span> <span m="1518770">the</span> <span m="1519000">order</span> <span m="1521920">of</span> <span m="1522020">the</span> <span m="1522090">items</span> <span m="1523976">or</span> <span m="1524410">the</span> <span m="1524520">letters.</span> <span m="1529480">Proof--</span> <span m="1532790">well,</span> <span m="1533020">it's</span> <span m="1533160">obvious.</span> <span m="1539540">No,</span> <span m="1539720">you</span> <span m="1539850">can't</span> <span m="1540050">let</span> <span m="1540140">me</span> <span m="1540240">get</span> <span m="1540370">away</span> <span m="1540430">with</span> <span m="1540670">that,</span> <span m="1540910">right?</span> <span m="1541640">Because</span> <span m="1541810">I'm</span> <span m="1541940">not</span> <span m="1542110">to</span> <span m="1542200">let</span> <span m="1542350">you</span> <span m="1542520">get</span> <span m="1542670">away</span> <span m="1542860">with</span> <span m="1542980">that</span> <span m="1543900">on</span> <span m="1544050">homework.</span> <span m="1544490">That's</span> <span m="1544740">not</span> <span m="1544930">so</span> <span m="1545060">good.</span> </p>
<p><span m="1547120">So</span> <span m="1547270">we've</span> <span m="1547330">got</span> <span m="1547470">to</span> <span m="1547510">be</span> <span m="1547590">a</span> <span m="1547640">little</span> <span m="1547890">more</span> <span m="1548070">careful</span> <span m="1549660">in</span> <span m="1549780">the</span> <span m="1549880">proof</span> <span m="1550210">here.</span> <span m="1551140">Otherwise,</span> <span m="1551650">we</span> <span m="1551770">just</span> <span m="1552100">head</span> <span m="1552270">down</span> <span m="1552440">the</span> <span m="1552520">path</span> <span m="1552790">to</span> <span m="1552880">trouble.</span> <span m="1554580">So</span> <span m="1554720">let's</span> <span m="1554910">sort</span> <span m="1555080">of</span> <span m="1555560">specify,</span> <span m="1556370">what</span> <span m="1556660">does</span> <span m="1556810">a</span> <span m="1556950">row</span> <span m="1557210">move</span> <span m="1557520">mean--</span> <span m="1558320">just</span> <span m="1558850">get</span> <span m="1559760">a</span> <span m="1560160">little</span> <span m="1560320">bit</span> <span m="1560450">of</span> <span m="1560510">math</span> <span m="1560900">and</span> <span m="1562750">specification</span> <span m="1563450">around</span> <span m="1563730">it.</span> </p>
<p><span m="1565140">Well,</span> <span m="1565890">in</span> <span m="1566030">a</span> <span m="1566140">row</span> <span m="1566360">move,</span> <span m="1570600">we</span> <span m="1570850">move</span> <span m="1572940">an</span> <span m="1573210">item--</span> <span m="1573720">a</span> <span m="1573820">letter--</span> <span m="1579410">from</span> <span m="1581030">some</span> <span m="1581340">cell</span> <span m="1581870">i,</span> <span m="1583360">whatever</span> <span m="1583680">i--</span> <span m="1583920">it</span> <span m="1584140">could</span> <span m="1584250">be</span> <span m="1584360">anything</span> <span m="1585670">1 to</span> <span m="1585910">9--</span> <span m="1587280">into</span> <span m="1588940">an</span> <span m="1589220">adjacent</span> <span m="1589840">cell.</span> <span m="1595960">What</span> <span m="1596460">are</span> <span m="1596570">the</span> <span m="1596680">possibilities</span> <span m="1597400">for</span> <span m="1597510">the</span> <span m="1597630">adjacent</span> <span m="1598140">cell?</span> <span m="1600720">Plus</span> <span m="1600950">or</span> <span m="1601000">minus</span> <span m="1601320">1,</span> <span m="1601580">yeah,</span> <span m="1603580">into</span> <span m="1603860">cell</span> <span m="1604990">i</span> <span m="1605360">minus</span> <span m="1605760">1</span> <span m="1606240">or</span> <span m="1606760">i</span> <span m="1607110">plus</span> <span m="1607450">1.</span> <span m="1610120">Nothing</span> <span m="1610570">else</span> <span m="1610810">moves.</span> </p>
<p></p>
<p><span m="1619230">Hence,</span> <span m="1622360">the</span> <span m="1622550">order</span> <span m="1624780">of</span> <span m="1624920">the</span> <span m="1625030">items</span> <span m="1625370">is</span> <span m="1625480">preserved.</span> </p>
<p></p>
<p><span m="1636640">Because</span> <span m="1637110">if</span> <span m="1637180">you're</span> <span m="1637270">going</span> <span m="1637750">with</span> <span m="1638130">i to</span> <span m="1638320">i</span> <span m="1638660">plus</span> <span m="1638970">1,</span> <span m="1639600">everything</span> <span m="1639980">else</span> <span m="1640230">is</span> <span m="1640350">in</span> <span m="1640480">i</span> <span m="1640590">minus</span> <span m="1640930">1</span> <span m="1641060">or</span> <span m="1641110">less,</span> <span m="1641800">i</span> <span m="1641940">plus</span> <span m="1642240">2</span> <span m="1642410">or</span> <span m="1642460">more.</span> <span m="1643450">Relative</span> <span m="1643860">order</span> <span m="1644050">does</span> <span m="1644210">not</span> <span m="1644410">change</span> <span m="1645140">in</span> <span m="1645240">this</span> <span m="1645420">case.</span> <span m="1648450">And</span> <span m="1648610">I'll</span> <span m="1648650">claim</span> <span m="1648940">that's</span> <span m="1649180">enough.</span> </p>
<p><span m="1650570">You</span> <span m="1651430">know</span> <span m="1651690">that</span> <span m="1651830">I've</span> <span m="1651960">thought</span> <span m="1652210">about</span> <span m="1652460">it a</span> <span m="1652510">little</span> <span m="1652720">bit.</span> <span m="1652910">I've</span> <span m="1653030">really</span> <span m="1653240">quantified</span> <span m="1653790">that</span> <span m="1653970">i</span> <span m="1654240">goes</span> <span m="1654530">to i</span> <span m="1654650">minus</span> <span m="1655010">1 or</span> <span m="1655210">i</span> <span m="1655280">plus</span> <span m="1655580">1.</span> <span m="1655820">Nothing</span> <span m="1656100">else</span> <span m="1656290">moves.</span> <span m="1657460">Therefore,</span> <span m="1658300">we're</span> <span m="1658430">done.</span> <span m="1658780">Now, I</span> <span m="1658890">could've</span> <span m="1659690">added</span> <span m="1660010">some</span> <span m="1660130">more</span> <span m="1660350">sentences</span> <span m="1660780">here,</span> <span m="1660920">like</span> <span m="1661110">I</span> <span m="1661170">talked</span> <span m="1661520">about.</span> <span m="1662400">But</span> <span m="1662450">this</span> <span m="1662620">is</span> <span m="1662730">probably</span> <span m="1663180">enough</span> <span m="1663520">in</span> <span m="1663600">this</span> <span m="1663800">case,</span> <span m="1665480">for</span> <span m="1665590">this</span> <span m="1665810">guy.</span> </p>
<p><span m="1668610">So</span> <span m="1670580">in</span> <span m="1670910">fact,</span> <span m="1671730">if</span> <span m="1671780">we</span> <span m="1671920">only</span> <span m="1672180">had</span> <span m="1672350">row</span> <span m="1672610">moves,</span> <span m="1673510">there's</span> <span m="1673640">not</span> <span m="1673830">much</span> <span m="1674030">that</span> <span m="1674140">happens.</span> <span m="1674650">The</span> <span m="1674670">order</span> <span m="1675080">never</span> <span m="1675410">changes</span> <span m="1675920">with</span> <span m="1676090">row</span> <span m="1676280">moves.</span> <span m="1677270">So</span> <span m="1677390">we</span> <span m="1677460">never</span> <span m="1677670">get</span> <span m="1677870">there.</span> <span m="1678100">But</span> <span m="1678270">column</span> <span m="1678790">moves</span> <span m="1678940">are</span> <span m="1678990">more</span> <span m="1679200">interesting.</span> </p>
<p></p>
<p><span m="1686320">So</span> <span m="1686550">as</span> <span m="1686650">an</span> <span m="1686740">example</span> <span m="1687500">of</span> <span m="1688090">a</span> <span m="1688140">column</span> <span m="1688570">move,</span> <span m="1693240">a,</span> <span m="1693750">b,</span> <span m="1694620">c,</span> <span m="1695490">d,</span> <span m="1696360">f,</span> <span m="1697701">h,</span> <span m="1698595">e,</span> <span m="1699042">g--</span> <span m="1700000">say</span> <span m="1700340">I</span> <span m="1700380">move</span> <span m="1700770">the g</span> <span m="1700870">up.</span> <span m="1703220">Then</span> <span m="1703430">I</span> <span m="1703510">would</span> <span m="1703680">get</span> <span m="1705590">this</span> <span m="1705810">one--</span> <span m="1708850">a,</span> <span m="1709236">b, c,</span> <span m="1710780">d,</span> <span m="1711227">f,</span> <span m="1712121">g--</span> <span m="1712570">because</span> <span m="1712810">g</span> <span m="1713000">moved</span> <span m="1713250">up--</span> <span m="1714150">h, e.</span> <span m="1715420">So</span> <span m="1715600">g</span> <span m="1715920">went to</span> <span m="1716190">there.</span> </p>
<p><span m="1720240">Did</span> <span m="1720380">the</span> <span m="1720500">ordering</span> <span m="1720880">change</span> <span m="1722240">for</span> <span m="1722340">this</span> <span m="1722530">move?</span> <span m="1723690">Yeah.</span> <span m="1726300">How</span> <span m="1726470">so?</span> <span m="1728450">What</span> <span m="1728660">things</span> <span m="1728950">changed</span> <span m="1729350">order?</span> <span m="1729570">Which</span> <span m="1730180">pairs</span> <span m="1730470">of</span> <span m="1730570">letters</span> <span m="1730900">changed</span> <span m="1731260">order?</span> </p>
<p><span m="1734576">AUDIENCE: G changed</span> <span m="1735067">[INAUDIBLE].</span> </p>
<p><span m="1736540">PROFESSOR: G</span> <span m="1736970">changed</span> <span m="1737460">by</span> <span m="1737940">three--</span> <span m="1738970">moved</span> <span m="1739250">three</span> <span m="1739540">in</span> <span m="1739620">the</span> <span m="1739730">order.</span> <span m="1740450">And</span> <span m="1740750">what</span> <span m="1740930">letters</span> <span m="1741370">did</span> <span m="1741510">it</span> <span m="1741950">change</span> <span m="1742350">relative</span> <span m="1742810">order</span> <span m="1743030">with?</span> </p>
<p><span m="1744837">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="1747831">AUDIENCE: The two</span> <span m="1748330">preceding it.</span> </p>
<p><span m="1749830">PROFESSOR: Yeah,</span> <span m="1750040">the two</span> <span m="1750650">preceding</span> <span m="1751230">it.</span> <span m="1751800">g</span> <span m="1752150">used</span> <span m="1752430">to</span> <span m="1752480">be</span> <span m="1752700">after</span> <span m="1753220">h</span> <span m="1753470">and</span> <span m="1753540">e.</span> <span m="1754790">Now</span> <span m="1755480">it's</span> <span m="1755690">before</span> <span m="1756260">h and</span> <span m="1756750">e.</span> <span m="1758070">All</span> <span m="1758210">right,</span> <span m="1758520">so</span> <span m="1758680">it</span> <span m="1759220">changed</span> <span m="1760360">relative</span> <span m="1760840">order</span> <span m="1761060">with</span> <span m="1761210">two</span> <span m="1761400">items.</span> <span m="1761750">It</span> <span m="1761870">changed</span> <span m="1762220">its</span> <span m="1762330">position</span> <span m="1762780">by</span> <span m="1762920">three.</span> <span m="1763300">It moved</span> <span m="1763580">up</span> <span m="1763760">three</span> <span m="1764340">in</span> <span m="1764430">the</span> <span m="1764530">ordering.</span> <span m="1765100">And</span> <span m="1765220">so</span> <span m="1765350">it</span> <span m="1765410">changed</span> <span m="1765820">order</span> <span m="1765970">with</span> <span m="1766140">the</span> <span m="1766200">two</span> <span m="1766550">in</span> <span m="1766630">between.</span> </p>
<p><span m="1768020">Let's</span> <span m="1768200">look</span> <span m="1768330">at</span> <span m="1768370">another</span> <span m="1768650">one.</span> <span m="1770630">Because</span> <span m="1770820">we're</span> <span m="1771030">really</span> <span m="1771330">doing</span> <span m="1772110">these</span> <span m="1772270">examples</span> <span m="1772760">to</span> <span m="1772850">hunt</span> <span m="1773240">for</span> <span m="1773390">an</span> <span m="1773470">invariant,</span> <span m="1774870">is</span> <span m="1775050">what</span> <span m="1775190">we're</span> <span m="1775300">trying</span> <span m="1775560">to</span> <span m="1775620">do</span> <span m="1775810">here--</span> <span m="1777130">something</span> <span m="1777530">that will</span> <span m="1777650">let us</span> <span m="1778200">prove</span> <span m="1778500">we</span> <span m="1778610">can</span> <span m="1778780">never</span> <span m="1779055">get</span> <span m="1779330">to</span> <span m="1779420">the</span> <span m="1779490">desired</span> <span m="1779930">state.</span> <span m="1781710">Let's</span> <span m="1781840">look</span> <span m="1781960">at</span> <span m="1782040">this</span> <span m="1782260">one.</span> </p>
<p><span m="1783750">a,</span> <span m="1784010">b,</span> <span m="1784726">c,</span> <span m="1785638">d,</span> <span m="1786550">g,</span> <span m="1787720">h,</span> <span m="1788150">e,</span> <span m="1788580">f</span> <span m="1789800">goes</span> <span m="1790210">to--</span> <span m="1794010">I'm</span> <span m="1794050">going</span> <span m="1794140">to</span> <span m="1794180">move</span> <span m="1794400">b</span> <span m="1794640">down</span> <span m="1794990">this</span> <span m="1795160">time--</span> <span m="1797640">a,</span> <span m="1798060">c,</span> <span m="1799240">d,</span> <span m="1799676">b,</span> <span m="1800548">g,</span> <span m="1802130">h,</span> <span m="1802962">e,</span> <span m="1803378">f.</span> <span m="1805680">So</span> <span m="1805880">b</span> <span m="1806300">moves</span> <span m="1806650">down.</span> <span m="1809225">The</span> <span m="1809650">relative</span> <span m="1810060">order</span> <span m="1810270">changes.</span> <span m="1812470">Which</span> <span m="1812750">pairs</span> <span m="1813500">got</span> <span m="1813700">changed</span> <span m="1814250">in</span> <span m="1814310">the</span> <span m="1814440">relative</span> <span m="1814780">order?</span> </p>
<p><span m="1819870">AUDIENCE: b.</span> </p>
<p><span m="1820820">PROFESSOR: b</span> <span m="1821310">changed</span> <span m="1821810">with--</span> </p>
<p><span m="1822490">AUDIENCE: c and</span> <span m="1822960">[INAUDIBLE].</span> </p>
<p><span m="1823430">PROFESSOR: c</span> <span m="1823770">and</span> <span m="1826190">d.</span> <span m="1827070">b</span> <span m="1827510">used</span> <span m="1827720">to</span> <span m="1827760">be</span> <span m="1827860">before</span> <span m="1828260">c</span> <span m="1828510">and</span> <span m="1828610">d.</span> <span m="1829000">when it</span> <span m="1829120">moves</span> <span m="1829490">down</span> <span m="1829640">here,</span> <span m="1830380">it</span> <span m="1830500">goes</span> <span m="1830730">after</span> <span m="1831160">c</span> <span m="1831455">and</span> <span m="1831750">d.</span> <span m="1834160">Any</span> <span m="1834350">guess</span> <span m="1834670">about</span> <span m="1835000">what</span> <span m="1835140">Lemma</span> <span m="1835410">Two</span> <span m="1835590">is</span> <span m="1835700">going</span> <span m="1835810">to</span> <span m="1835870">be?</span> <span m="1837580">What</span> <span m="1837740">happens</span> <span m="1838140">when</span> <span m="1838220">we</span> <span m="1838330">make</span> <span m="1838610">a</span> <span m="1838720">column</span> <span m="1839110">move?</span> </p>
<p><span m="1840652">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="1841460">PROFESSOR: Change</span> <span m="1841800">the</span> <span m="1841930">order</span> <span m="1842520">of--</span> </p>
<p><span m="1844616">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="1850030">PROFESSOR: Good,</span> <span m="1850860">so</span> <span m="1851690">a</span> <span m="1851760">column</span> <span m="1852160">move</span> <span m="1852770">changes</span> <span m="1853210">the</span> <span m="1853360">order</span> <span m="1854140">of</span> <span m="1854700">a</span> <span m="1854810">guy</span> <span m="1855060">with</span> <span m="1855200">the</span> <span m="1855290">previous</span> <span m="1855680">two</span> <span m="1855980">or</span> <span m="1856230">the</span> <span m="1856340">next</span> <span m="1856630">two,</span> <span m="1856770">which</span> <span m="1856940">means</span> <span m="1857300">the</span> <span m="1857380">relative</span> <span m="1857730">order</span> <span m="1857910">of</span> <span m="1858000">two</span> <span m="1858140">pairs</span> <span m="1858470">change.</span> <span m="1859920">So</span> <span m="1860060">let's</span> <span m="1860230">take</span> <span m="1860410">that</span> <span m="1860570">as Lemma</span> <span m="1860910">Two.</span> <span m="1864750">When</span> <span m="1864850">we</span> <span m="1864960">say</span> <span m="1865240">don't</span> <span m="1865500">do</span> <span m="1865630">proof</span> <span m="1865920">by</span> <span m="1866080">example,</span> <span m="1866660">we</span> <span m="1866670">don't</span> <span m="1866920">mean</span> <span m="1867590">don't</span> <span m="1867830">try</span> <span m="1868060">examples.</span> <span m="1869220">Because</span> <span m="1869710">by</span> <span m="1869800">trying</span> <span m="1870050">examples,</span> <span m="1870530">you</span> <span m="1870620">find</span> <span m="1870890">out</span> <span m="1871300">what</span> <span m="1871490">you're</span> <span m="1871570">trying</span> <span m="1871790">to</span> <span m="1871850">prove.</span> <span m="1873370">So</span> <span m="1873390">that's</span> <span m="1873620">good</span> <span m="1873750">to</span> <span m="1873830">do.</span> </p>
<p><span m="1875680">A</span> <span m="1875810">column</span> <span m="1876280">move</span> <span m="1881910">changes</span> <span m="1885150">the</span> <span m="1886070">relative</span> <span m="1886500">order</span> <span m="1893820">of</span> <span m="1894250">precisely</span> <span m="1898580">two</span> <span m="1899600">pairs</span> <span m="1900100">of</span> <span m="1900210">items.</span> <span m="1906860">So</span> <span m="1906920">let's</span> <span m="1907100">prove</span> <span m="1907400">that.</span> <span m="1913130">In</span> <span m="1913300">a</span> <span m="1913360">column</span> <span m="1913820">move,</span> <span m="1920320">we</span> <span m="1920470">move</span> <span m="1920750">an</span> <span m="1920860">item</span> <span m="1921210">in</span> <span m="1921290">cell</span> <span m="1921610">i</span> <span m="1922620">for</span> <span m="1922750">some</span> <span m="1922970">i</span> <span m="1932010">to</span> <span m="1932460">a</span> <span m="1932510">blank</span> <span m="1932870">spot</span> <span m="1938980">in</span> <span m="1939470">cell</span> <span m="1940770">what?</span> </p>
<p><span m="1942642">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="1944580">PROFESSOR: i</span> <span m="1944790">minus</span> <span m="1945140">3 or i</span> <span m="1945540">plus</span> <span m="1945860">3?</span> </p>
<p><span m="1948210">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="1951860">PROFESSOR: OK,</span> <span m="1952760">and</span> <span m="1953000">just</span> <span m="1953180">to</span> <span m="1953250">see</span> <span m="1953560">that,</span> <span m="1955050">let's</span> <span m="1955260">draw</span> <span m="1955560">out</span> <span m="1955640">the</span> <span m="1955710">natural</span> <span m="1956080">order.</span> <span m="1956340">I've</span> <span m="1956470">got 1</span> <span m="1956924">2,</span> <span m="1957378">3,</span> <span m="1957832">4, 5,</span> <span m="1958286">6,</span> <span m="1958740">7,</span> <span m="1959194">8,</span> <span m="1959648">9.</span> <span m="1960890">Column</span> <span m="1961380">moves</span> <span m="1961810">could</span> <span m="1961900">be</span> <span m="1962060">here--</span> <span m="1963430">1</span> <span m="1963690">to</span> <span m="1963770">4,</span> <span m="1964150">and</span> <span m="1964270">4</span> <span m="1964470">to</span> <span m="1964550">1,</span> <span m="1965700">2</span> <span m="1965910">and</span> <span m="1966040">5,</span> <span m="1966690">3</span> <span m="1967030">and</span> <span m="1967150">6,</span> <span m="1967830">4</span> <span m="1968120">and</span> <span m="1968230">7,</span> <span m="1969480">5</span> <span m="1969750">and</span> <span m="1969840">8,</span> <span m="1970245">6 and</span> <span m="1970650">9.</span> </p>
<p><span m="1971880">All</span> <span m="1972280">12</span> <span m="1972740">possible</span> <span m="1973200">column</span> <span m="1973630">moves,</span> <span m="1974940">this</span> <span m="1975140">works</span> <span m="1975430">for.</span> <span m="1975810">They're</span> <span m="1975970">always</span> <span m="1976370">separated</span> <span m="1976760">by</span> <span m="1976880">3.</span> <span m="1980170">And</span> <span m="1981040">when</span> <span m="1982180">an</span> <span m="1982360">item</span> <span m="1982770">moves</span> <span m="1983050">three</span> <span m="1983260">positions,</span> <span m="1997080">it</span> <span m="1997290">changes</span> <span m="1997730">relative</span> <span m="1998190">order</span> <span m="1998490">with</span> <span m="1998640">two</span> <span m="1998870">other</span> <span m="1999080">items.</span> </p>
<p><span m="2007360">And</span> <span m="2007480">you</span> <span m="2007590">can</span> <span m="2007710">even</span> <span m="2007870">write</span> <span m="2008100">down</span> <span m="2008270">what</span> <span m="2008400">they</span> <span m="2008550">are.</span> <span m="2009950">It's</span> <span m="2010760">i</span> <span m="2010910">minus</span> <span m="2011290">1,</span> <span m="2011690">i</span> <span m="2011820">minus</span> <span m="2012190">2,</span> <span m="2013136">or</span> <span m="2013990">i</span> <span m="2014240">plus</span> <span m="2014580">1,</span> <span m="2015040">i</span> <span m="2015220">plus</span> <span m="2015630">2.</span> <span m="2016980">Those</span> <span m="2017240">are</span> <span m="2017270">the</span> <span m="2017340">guys</span> <span m="2017560">it</span> <span m="2017640">changes</span> <span m="2018000">order</span> <span m="2018330">with.</span> </p>
<p><span m="2022560">All right, we've</span> <span m="2022830">got</span> <span m="2023020">two</span> <span m="2023190">lemmas</span> <span m="2023540">here.</span> <span m="2024100">That</span> <span m="2024480">one's</span> <span m="2024660">done</span> <span m="2024870">now.</span> <span m="2026380">In</span> <span m="2026490">a</span> <span m="2026590">row</span> <span m="2026860">move,</span> <span m="2027500">order</span> <span m="2027770">does</span> <span m="2027960">not</span> <span m="2028180">change.</span> <span m="2029100">In</span> <span m="2029200">a</span> <span m="2029280">column</span> <span m="2029770">move,</span> <span m="2030900">two</span> <span m="2031210">pairs</span> <span m="2031690">of</span> <span m="2031810">letters</span> <span m="2032860">get</span> <span m="2033070">flipped.</span> </p>
<p><span m="2035070">That's</span> <span m="2035300">all</span> <span m="2035500">you</span> <span m="2035630">can</span> <span m="2035780">do</span> <span m="2035910">with</span> <span m="2036010">this</span> <span m="2036160">puzzle</span> <span m="2036450">now.</span> <span m="2038240">So</span> <span m="2038430">can</span> <span m="2038580">anybody</span> <span m="2039080">think</span> <span m="2039400">of</span> <span m="2041420">something</span> <span m="2041950">we</span> <span m="2042050">can</span> <span m="2042190">work</span> <span m="2042480">with</span> <span m="2042610">to</span> <span m="2042670">get</span> <span m="2042830">an</span> <span m="2042910">invariant</span> <span m="2043440">here--</span> <span m="2044560">something</span> <span m="2044890">that</span> <span m="2044990">won't</span> <span m="2045220">change?</span> <span m="2046750">What</span> <span m="2046870">we</span> <span m="2046950">should</span> <span m="2047050">be</span> <span m="2047170">focusing</span> <span m="2047590">on?</span> <span m="2047710">Yeah.</span> </p>
<p><span m="2049070">AUDIENCE: You can only</span> <span m="2050540">switch</span> <span m="2051030">two pairs</span> <span m="2051520">of items</span> <span m="2052010">at a</span> <span m="2052500">time.</span> <span m="2052990">So if</span> <span m="2054460">you don't</span> <span m="2054950">have an even</span> <span m="2055440">number</span> <span m="2055930">[INAUDIBLE]</span> <span m="2058380">switch</span> <span m="2058870">[INAUDIBLE].</span> </p>
<p><span m="2060850">PROFESSOR: Yeah,</span> <span m="2061580">that's</span> <span m="2061929">good.</span> <span m="2062199">So</span> <span m="2062760">really,</span> <span m="2063050">to</span> <span m="2063120">focus</span> <span m="2063570">in</span> <span m="2063690">on</span> <span m="2063820">there</span> <span m="2064040">is,</span> <span m="2064139">if</span> <span m="2064219">you</span> <span m="2064320">have</span> <span m="2064670">an</span> <span m="2065199">even</span> <span m="2065560">number</span> <span m="2065929">of</span> <span m="2066040">things</span> <span m="2066480">out</span> <span m="2066639">of</span> <span m="2066739">order</span> <span m="2067600">to</span> <span m="2067710">start,</span> <span m="2069040">you're</span> <span m="2069179">going</span> <span m="2069280">to</span> <span m="2069320">have</span> <span m="2069489">an</span> <span m="2069570">even</span> <span m="2069800">number</span> <span m="2070199">forever,</span> <span m="2070620">because</span> <span m="2070780">you</span> <span m="2070870">can</span> <span m="2070949">only</span> <span m="2071130">change</span> <span m="2071440">two</span> <span m="2071630">at</span> <span m="2071679">a</span> <span m="2071739">time.</span> <span m="2072860">That's</span> <span m="2073040">a</span> <span m="2073100">great</span> <span m="2073460">idea.</span> <span m="2073820">Let's</span> <span m="2074630">really</span> <span m="2074980">specify</span> <span m="2075590">that</span> <span m="2075860">now,</span> <span m="2077100">and</span> <span m="2077510">define</span> <span m="2077920">that.</span> </p>
<p><span m="2082840">A</span> <span m="2082940">pair</span> <span m="2083330">of</span> <span m="2083440">letters</span> <span m="2085469">or</span> <span m="2085670">items,</span> <span m="2087909">call them</span> <span m="2088239">L1</span> <span m="2089600">and</span> <span m="2090090">L2--</span> <span m="2092420">they</span> <span m="2092580">form</span> <span m="2092909">an</span> <span m="2092980">inversion,</span> <span m="2103270">also</span> <span m="2103610">known</span> <span m="2103880">as</span> <span m="2104340">an</span> <span m="2104410">inverted</span> <span m="2104860">pair,</span> <span m="2112520">if</span> <span m="2113140">L1</span> <span m="2114520">precedes</span> <span m="2117730">L2</span> <span m="2118410">in</span> <span m="2118480">the</span> <span m="2118600">alphabet,</span> <span m="2124160">but</span> <span m="2124420">L1</span> <span m="2125230">appears</span> <span m="2125630">after</span> <span m="2126110">L2</span> <span m="2126540">in</span> <span m="2126600">the</span> <span m="2126680">puzzle.</span> <span m="2138170">All</span> <span m="2138280">right,</span> <span m="2138450">so</span> <span m="2139150">that</span> <span m="2139490">is</span> <span m="2139620">an</span> <span m="2139700">inversion.</span> <span m="2141470">So</span> <span m="2141710">for</span> <span m="2141830">example,</span> <span m="2142880">let's</span> <span m="2143030">see</span> <span m="2143150">how</span> <span m="2143290">many</span> <span m="2143540">inversions</span> <span m="2143960">do</span> <span m="2144040">we</span> <span m="2144140">have</span> <span m="2144380">in</span> <span m="2144440">this</span> <span m="2144630">case?</span> </p>
<p><span m="2149060">We've got</span> <span m="2149360">a,</span> <span m="2149930">b,</span> <span m="2150406">c,</span> <span m="2151834">f,</span> <span m="2152786">d,</span> <span m="2153740">g,</span> <span m="2155210">e,</span> <span m="2156190">h.</span> <span m="2157850">How</span> <span m="2158040">many</span> <span m="2158260">inversions</span> <span m="2158970">are</span> <span m="2159120">there</span> <span m="2159310">in</span> <span m="2159370">that</span> <span m="2159580">state</span> <span m="2160000">of</span> <span m="2160090">the</span> <span m="2160170">puzzle?</span> <span m="2172740">Yeah?</span> </p>
<p><span m="2174070">AUDIENCE: Three.</span> </p>
<p><span m="2174950">PROFESSOR: Three, good.</span> <span m="2175600">And</span> <span m="2176000">what</span> <span m="2176200">are</span> <span m="2176340">they?</span> </p>
<p><span m="2177866">AUDIENCE: d</span> <span m="2178318">is</span> <span m="2178770">after</span> <span m="2179222">f.</span> </p>
<p><span m="2180130">PROFESSOR: Yeah,</span> <span m="2180830">d,</span> <span m="2181155">f</span> <span m="2181480">is</span> <span m="2181870">an</span> <span m="2181910">inversion.</span> <span m="2186340">e,</span> <span m="2186680">f</span> <span m="2187020">is</span> <span m="2187140">an</span> <span m="2187230">inversion.</span> <span m="2189600">And</span> <span m="2190760">what's</span> <span m="2190920">the</span> <span m="2190980">last</span> <span m="2191220">one?</span> </p>
<p><span m="2191480">AUDIENCE: e, g.</span> </p>
<p><span m="2192250">PROFESSOR: e, g</span> <span m="2192720">is</span> <span m="2192820">an</span> <span m="2192860">inversion.</span> <span m="2194320">Yes,</span> <span m="2194720">so</span> <span m="2194860">the</span> <span m="2195000">answer is</span> <span m="2195330">there's</span> <span m="2195580">three</span> <span m="2195910">inversions</span> <span m="2197430">in</span> <span m="2197590">this</span> <span m="2197800">puzzle.</span> <span m="2199680">And</span> <span m="2200090">if</span> <span m="2200210">I</span> <span m="2200340">keep</span> <span m="2200700">doing</span> <span m="2201230">row</span> <span m="2201490">and</span> <span m="2201570">column</span> <span m="2201960">moves</span> <span m="2202240">to</span> <span m="2202340">this,</span> <span m="2203800">what</span> <span m="2203980">do</span> <span m="2204070">I</span> <span m="2204150">know</span> <span m="2204360">about</span> <span m="2204600">the</span> <span m="2204710">parity</span> <span m="2205200">of</span> <span m="2205270">the</span> <span m="2205340">number</span> <span m="2205730">of</span> <span m="2205900">inversions?</span> </p>
<p><span m="2207746">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="2210430">PROFESSOR: Will</span> <span m="2210660">always</span> <span m="2211180">be</span> <span m="2211380">odd--</span> <span m="2212130">that's</span> <span m="2212430">sort</span> <span m="2212620">of</span> <span m="2212710">interesting.</span> <span m="2214040">OK,</span> <span m="2216830">now,</span> <span m="2217410">how</span> <span m="2217670">many</span> <span m="2217900">inverted</span> <span m="2218300">pairs</span> <span m="2218850">are</span> <span m="2219150">there</span> <span m="2220260">in</span> <span m="2220570">the</span> <span m="2220670">start</span> <span m="2221240">state</span> <span m="2222180">way</span> <span m="2222420">back</span> <span m="2222710">over</span> <span m="2222960">here?</span> <span m="2225100">How</span> <span m="2225260">many</span> <span m="2225460">inverted</span> <span m="2225840">pairs</span> <span m="2226220">here?</span> <span m="2228440">One</span> <span m="2228840">inversion.</span> </p>
<p><span m="2234120">How</span> <span m="2234360">many</span> <span m="2235170">inversions</span> <span m="2235860">here?</span> <span m="2238080">Zero.</span> <span m="2241960">All</span> <span m="2242220">right,</span> <span m="2243310">and</span> <span m="2243560">can</span> <span m="2243690">you</span> <span m="2243780">see</span> <span m="2244060">where</span> <span m="2244190">we're</span> <span m="2244290">going</span> <span m="2244410">to</span> <span m="2244480">head</span> <span m="2244670">here</span> <span m="2244970">now?</span> <span m="2247350">If I</span> <span m="2247470">started</span> <span m="2247880">with</span> <span m="2248040">one,</span> <span m="2250380">that's</span> <span m="2250630">odd.</span> <span m="2252080">And</span> <span m="2252270">every</span> <span m="2252490">time</span> <span m="2252750">I</span> <span m="2252880">do</span> <span m="2253150">a</span> <span m="2253230">row move or</span> <span m="2253650">column</span> <span m="2254080">move,</span> <span m="2254190">it's going</span> <span m="2254290">to</span> <span m="2254340">stay</span> <span m="2254610">odd.</span> <span m="2255750">And</span> <span m="2255890">I</span> <span m="2255960">can</span> <span m="2256270">never</span> <span m="2256630">get</span> <span m="2256870">here.</span> </p>
<p><span m="2258930">Is that</span> <span m="2259060">clear</span> <span m="2259540">what</span> <span m="2259720">we're</span> <span m="2260230">trying</span> <span m="2260470">to</span> <span m="2260540">do?</span> <span m="2261190">So</span> <span m="2261340">let's</span> <span m="2261590">keep</span> <span m="2261780">doing</span> <span m="2262050">it.</span> <span m="2266550">Let's</span> <span m="2266840">go</span> <span m="2266950">over</span> <span m="2267070">here.</span> </p>
<p><span m="2279110">All</span> <span m="2279180">right,</span> <span m="2279350">so</span> <span m="2279480">now</span> <span m="2279790">we'll</span> <span m="2280880">use</span> <span m="2281220">Lemma One and</span> <span m="2281650">Lemma</span> <span m="2281910">Two</span> <span m="2282510">create</span> <span m="2282830">this</span> <span m="2283020">notion</span> <span m="2283380">that</span> <span m="2283500">we</span> <span m="2284410">only</span> <span m="2284700">change</span> <span m="2285020">the</span> <span m="2285140">number of</span> <span m="2285340">inversions</span> <span m="2285820">by</span> <span m="2286630">an</span> <span m="2286720">even</span> <span m="2286930">number.</span> <span m="2287883">And</span> <span m="2288336">Lemma</span> <span m="2288790">Three--</span> <span m="2293630">during</span> <span m="2293930">a</span> <span m="2293980">move,</span> <span m="2299040">the</span> <span m="2299260">number</span> <span m="2300950">of</span> <span m="2301150">inversions</span> <span m="2308130">can</span> <span m="2308380">only</span> <span m="2311130">increase</span> <span m="2311650">by</span> <span m="2311790">two,</span> <span m="2316680">decrease</span> <span m="2317130">by</span> <span m="2317280">two,</span> <span m="2322500">or</span> <span m="2323440">stay</span> <span m="2323620">the</span> <span m="2323750">same.</span> </p>
<p><span m="2329510">And</span> <span m="2329620">this</span> <span m="2329770">proof</span> <span m="2329970">is</span> <span m="2330080">pretty</span> <span m="2330370">easy.</span> <span m="2331210">There's</span> <span m="2331780">a</span> <span m="2332470">couple</span> <span m="2332710">of</span> <span m="2332760">cases.</span> <span m="2336910">In</span> <span m="2337040">a</span> <span m="2337160">row</span> <span m="2337430">move,</span> <span m="2339270">what</span> <span m="2339430">happens?</span> <span m="2341250">How</span> <span m="2342960">does</span> <span m="2343130">the</span> <span m="2343210">number</span> <span m="2343510">of inversions</span> <span m="2344380">change</span> <span m="2344790">in a row</span> <span m="2345290">move?</span> </p>
<p><span m="2345750">AUDIENCE: Stays the same.</span> </p>
<p><span m="2346670">PROFESSOR: Stays</span> <span m="2347050">the</span> <span m="2347140">same--</span> <span m="2347770">no</span> <span m="2348010">changes--</span> <span m="2349770">and</span> <span m="2350070">that's</span> <span m="2350290">by</span> <span m="2350420">Lemma</span> <span m="2350670">One.</span> <span m="2357910">Now,</span> <span m="2358080">in</span> <span m="2358150">a</span> <span m="2358210">column</span> <span m="2358660">move,</span> <span m="2359950">there's</span> <span m="2360180">three</span> <span m="2360380">cases.</span> <span m="2366330">But</span> <span m="2366470">we</span> <span m="2366570">know</span> <span m="2367260">two</span> <span m="2367430">pairs</span> <span m="2367800">change</span> <span m="2368140">order--</span> <span m="2368340">that</span> <span m="2368560">we</span> <span m="2368660">know.</span> <span m="2369450">That's</span> <span m="2369700">Lemma</span> <span m="2369760">Two.</span> <span m="2374190">All</span> <span m="2374380">right,</span> <span m="2374580">by</span> <span m="2374770">Lemma</span> <span m="2374950">Two,</span> <span m="2375190">we</span> <span m="2375290">know</span> <span m="2375440">that.</span> <span m="2379600">But</span> <span m="2379740">the</span> <span m="2379820">three</span> <span m="2380060">cases</span> <span m="2380590">are--</span> <span m="2381270">let</span> <span m="2381350">me</span> <span m="2381480">start</span> <span m="2381710">on</span> <span m="2381860">the next</span> <span m="2382360">board.</span> </p>
<p></p>
<p><span m="2400780">So</span> <span m="2401030">Case</span> <span m="2401290">A</span> <span m="2401800">is</span> <span m="2401970">that</span> <span m="2403250">both</span> <span m="2403590">of</span> <span m="2403670">the</span> <span m="2403800">inverted</span> <span m="2404190">pairs</span> <span m="2407850">were</span> <span m="2408220">in</span> <span m="2408480">order</span> <span m="2409800">originally,</span> <span m="2410460">before</span> <span m="2410830">the</span> <span m="2410930">column</span> <span m="2411340">move.</span> <span m="2414030">What</span> <span m="2414300">happens</span> <span m="2414710">to</span> <span m="2414790">the</span> <span m="2414850">number</span> <span m="2415180">of</span> <span m="2415240">inversions</span> <span m="2415700">in</span> <span m="2415760">this</span> <span m="2415950">case,</span> <span m="2417720">as</span> <span m="2417890">a</span> <span m="2417940">result</span> <span m="2418210">of</span> <span m="2418250">the</span> <span m="2418330">column</span> <span m="2418700">move?</span> </p>
<p><span m="2420588">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="2422010">PROFESSOR: Goes</span> <span m="2422240">up</span> <span m="2422400">by</span> <span m="2422530">two.</span> <span m="2425700">Case</span> <span m="2426020">B</span> <span m="2427860">is</span> <span m="2428050">that</span> <span m="2428160">both</span> <span m="2428520">pairs</span> <span m="2429060">were</span> <span m="2429900">inverted</span> <span m="2431080">before</span> <span m="2431460">I</span> <span m="2431530">made</span> <span m="2431780">the</span> <span m="2431860">column</span> <span m="2432240">move.</span> <span m="2434250">What</span> <span m="2434420">happens</span> <span m="2434830">to</span> <span m="2434910">the</span> <span m="2434990">number</span> <span m="2435180">of</span> <span m="2435250">inversions</span> <span m="2435780">in</span> <span m="2435840">that</span> <span m="2436010">case?</span> </p>
<p><span m="2437994">AUDIENCE: Decreases by two.</span> </p>
<p><span m="2439430">PROFESSOR: Decreases</span> <span m="2439920">by</span> <span m="2440070">two--</span> <span m="2440270">they</span> <span m="2440410">were</span> <span m="2440560">in</span> <span m="2440690">order,</span> <span m="2441220">so</span> <span m="2441360">they</span> <span m="2441790">were</span> <span m="2441880">inverted.</span> <span m="2442700">Now</span> <span m="2443120">when</span> <span m="2443240">I</span> <span m="2443290">flop</span> <span m="2443630">them,</span> <span m="2443750">they</span> <span m="2443890">become</span> <span m="2445440">in</span> <span m="2445660">order.</span> <span m="2446010">So</span> <span m="2446520">inversions</span> <span m="2447060">drops</span> <span m="2447370">by</span> <span m="2447490">two.</span> </p>
<p><span m="2448470">And</span> <span m="2448760">Case C</span> <span m="2449450">is</span> <span m="2450030">there's</span> <span m="2450220">one</span> <span m="2450430">of</span> <span m="2450510">each.</span> <span m="2451650">One</span> <span m="2451910">of</span> <span m="2451950">the</span> <span m="2452050">pairs</span> <span m="2452390">was</span> <span m="2452520">inverted.</span> <span m="2457150">The</span> <span m="2457260">other</span> <span m="2457460">wasn't.</span> <span m="2458870">What</span> <span m="2459040">happens</span> <span m="2459450">to the</span> <span m="2459560">number</span> <span m="2459840">of</span> <span m="2459930">inversions</span> <span m="2460370">in</span> <span m="2460420">that</span> <span m="2460610">case?</span> </p>
<p><span m="2462040">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="2463030">PROFESSOR: Yes,</span> <span m="2463330">stays</span> <span m="2463610">the</span> <span m="2463700">same.</span> <span m="2470120">All right,</span> <span m="2470360">and</span> <span m="2470570">we're</span> <span m="2470640">done.</span> <span m="2471450">That's</span> <span m="2471650">what</span> <span m="2471740">the</span> <span m="2471820">lemma</span> <span m="2472160">says--</span> <span m="2472420">I</span> <span m="2472450">can</span> <span m="2472610">go</span> <span m="2472730">up</span> <span m="2473040">by two,</span> <span m="2473250">down</span> <span m="2473530">by</span> <span m="2473650">two,</span> <span m="2473880">or</span> <span m="2473960">stay</span> <span m="2474160">the</span> <span m="2474270">same.</span> </p>
<p><span m="2476990">All</span> <span m="2477240">right,</span> <span m="2477510">now</span> <span m="2479040">a</span> <span m="2479120">simple</span> <span m="2479400">corollary</span> <span m="2480410">of</span> <span m="2480570">this</span> <span m="2481820">is that</span> <span m="2482000">during</span> <span m="2482450">a</span> <span m="2482500">move,</span> <span m="2483260">the</span> <span m="2483440">parity--</span> <span m="2484470">i.e.</span> <span m="2484700">even</span> <span m="2485200">or</span> <span m="2485250">odd</span> <span m="2486200">of</span> <span m="2486590">the</span> <span m="2486790">number</span> <span m="2487100">of inversions</span> <span m="2488350">stays</span> <span m="2488890">the</span> <span m="2489000">same.</span> <span m="2489750">It</span> <span m="2489850">can't</span> <span m="2490140">change</span> <span m="2490450">at</span> <span m="2490510">all.</span> <span m="2499450">During</span> <span m="2499790">a</span> <span m="2499830">move,</span> <span m="2503290">the</span> <span m="2503400">parity--</span> <span m="2504250">and</span> <span m="2504440">the</span> <span m="2504520">parity</span> <span m="2504910">means</span> <span m="2505260">even,</span> <span m="2505540">odd.</span> <span m="2505840">Sometimes</span> <span m="2506190">it's</span> <span m="2506300">called</span> <span m="2506530">0,</span> <span m="2506870">1,</span> <span m="2510050">but</span> <span m="2510160">the</span> <span m="2510330">evenness</span> <span m="2510590">and</span> <span m="2510730">the</span> <span m="2510870">oddness</span> <span m="2512506">of</span> <span m="2512950">the</span> <span m="2513130">number</span> <span m="2513490">of</span> <span m="2513560">inversions</span> <span m="2520680">does</span> <span m="2520900">not</span> <span m="2521180">change.</span> </p>
<p><span m="2528456">AUDIENCE: Instead of two pairs</span> <span m="2528950">you mean</span> <span m="2529444">one pair,</span> <span m="2529938">right?</span> </p>
<p><span m="2531420">PROFESSOR: Where?</span> </p>
<p><span m="2532902">AUDIENCE: [INAUDIBLE]</span> <span m="2534806">pairs.</span> </p>
<p><span m="2535760">PROFESSOR: Oops,</span> <span m="2536520">I</span> <span m="2536650">mean</span> <span m="2536860">one</span> <span m="2537120">pair's</span> <span m="2537500">inverted,</span> <span m="2538530">one</span> <span m="2538750">pair's</span> <span m="2539090">not.</span> </p>
<p><span m="2540450">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="2543600">PROFESSOR: No,</span> <span m="2544050">I</span> <span m="2544160">mean</span> <span m="2544440">two</span> <span m="2544680">pairs</span> <span m="2545100">change</span> <span m="2545460">order</span> <span m="2545660">in</span> <span m="2545740">a</span> <span m="2545790">column</span> <span m="2546150">move.</span> <span m="2547860">Let's</span> <span m="2548010">go</span> <span m="2548100">back</span> <span m="2548300">and</span> <span m="2548370">look</span> <span m="2548530">at</span> <span m="2548590">that.</span> <span m="2550600">So</span> <span m="2550720">I'm</span> <span m="2550850">doing</span> <span m="2551050">a</span> <span m="2551110">column</span> <span m="2551550">move.</span> </p>
<p><span m="2551980">For</span> <span m="2552070">example,</span> <span m="2554040">g</span> <span m="2554310">goes</span> <span m="2554630">up.</span> <span m="2556490">g,</span> <span m="2556900">h</span> <span m="2557310">changes</span> <span m="2557730">order.</span> <span m="2558470">g,</span> <span m="2558795">e</span> <span m="2559120">changes</span> <span m="2559540">order.</span> <span m="2561320">g</span> <span m="2561650">changes</span> <span m="2562020">order</span> <span m="2562290">with</span> <span m="2562830">these</span> <span m="2563050">two</span> <span m="2563180">guys,</span> <span m="2563610">so</span> <span m="2563710">there's</span> <span m="2563790">two</span> <span m="2564030">pairs</span> <span m="2564590">that</span> <span m="2564850">got</span> <span m="2565180">flipped.</span> </p>
<p><span m="2567830">So</span> <span m="2568020">in</span> <span m="2568080">a</span> <span m="2568140">column</span> <span m="2568550">move,</span> <span m="2569070">two</span> <span m="2569320">pairs</span> <span m="2569860">reverse</span> <span m="2570300">their</span> <span m="2570500">order.</span> <span m="2571230">If they</span> <span m="2571400">were in</span> <span m="2571700">order,</span> <span m="2571890">they</span> <span m="2572010">become</span> <span m="2572270">inverted.</span> <span m="2573240">If</span> <span m="2573380">they</span> <span m="2573490">were</span> <span m="2573580">inverted,</span> <span m="2573920">they</span> <span m="2573990">become</span> <span m="2574290">in</span> <span m="2574380">order.</span> <span m="2576950">Any</span> <span m="2577110">other</span> <span m="2577260">questions?</span> <span m="2580390">Yeah.</span> </p>
<p><span m="2581350">AUDIENCE: So we're saying</span> <span m="2581830">that</span> <span m="2582310">letters</span> <span m="2582790">can be</span> <span m="2583270">numbers</span> <span m="2583750">in</span> <span m="2584230">more than</span> <span m="2584710">one pair?</span> </p>
<p><span m="2586160">PROFESSOR: Yes,</span> <span m="2586420">a</span> <span m="2586480">letter</span> <span m="2586780">can</span> <span m="2586930">be</span> <span m="2587070">in--</span> <span m="2587960">a</span> <span m="2588210">letter</span> <span m="2588470">is</span> <span m="2588760">in</span> <span m="2589630">seven</span> <span m="2589970">pairs,</span> <span m="2590820">because</span> <span m="2591150">there's</span> <span m="2591380">seven</span> <span m="2591590">other</span> <span m="2591750">letters.</span> <span m="2593250">And</span> <span m="2593410">in</span> <span m="2593480">fact,</span> <span m="2593710">we're</span> <span m="2593810">looking</span> <span m="2594110">at</span> <span m="2594170">a</span> <span m="2594230">space</span> <span m="2595240">of</span> <span m="2598190">8</span> <span m="2598400">times</span> <span m="2598650">7</span> <span m="2599000">divided</span> <span m="2599240">by</span> <span m="2599340">2--</span> <span m="2599823">28</span> <span m="2600790">pairs</span> <span m="2601810">of</span> <span m="2601930">letters.</span> <span m="2602195">So there</span> <span m="2602460">could</span> <span m="2602620">be</span> <span m="2602750">28</span> <span m="2603230">inversions</span> <span m="2603650">if</span> <span m="2603780">they're</span> <span m="2603920">all out of</span> <span m="2604250">order.</span> <span m="2606520">Any</span> <span m="2606690">other</span> <span m="2606830">questions?</span> </p>
<p><span m="2610600">OK,</span> <span m="2611350">well,</span> <span m="2611530">we're getting</span> <span m="2612310">there.</span> <span m="2612520">We're almost</span> <span m="2612780">done.</span> <span m="2612980">I've</span> <span m="2613020">got</span> <span m="2613130">to</span> <span m="2613190">prove</span> <span m="2613430">this</span> <span m="2613590">corollary</span> <span m="2613870">that</span> <span m="2614150">during</span> <span m="2614620">a</span> <span m="2614660">move,</span> <span m="2615720">the</span> <span m="2615950">evenness</span> <span m="2616650">or</span> <span m="2616880">oddness</span> <span m="2617290">of</span> <span m="2617360">the</span> <span m="2617430">number</span> <span m="2617640">of</span> <span m="2617710">inversions</span> <span m="2618210">does</span> <span m="2618330">not</span> <span m="2618530">change.</span> </p>
<p><span m="2622450">And</span> <span m="2622540">that's</span> <span m="2622770">because</span> <span m="2623740">adding</span> <span m="2624140">or</span> <span m="2624210">subtracting</span> <span m="2624900">2</span> <span m="2625160">from</span> <span m="2625350">a</span> <span m="2625400">number</span> <span m="2626410">does</span> <span m="2626560">not</span> <span m="2626810">change</span> <span m="2627230">its</span> <span m="2627350">parity.</span> <span m="2628440">It stays</span> <span m="2628830">odd or</span> <span m="2629100">stays</span> <span m="2629380">even.</span> <span m="2631780">So</span> <span m="2632040">adding</span> <span m="2633450">or</span> <span m="2633810">subtracting</span> <span m="2638720">2</span> <span m="2641560">does</span> <span m="2642120">not</span> <span m="2642480">change</span> <span m="2643030">the</span> <span m="2643130">parity.</span> </p>
<p><span m="2652170">All right,</span> <span m="2652440">now</span> <span m="2652770">we're</span> <span m="2652920">ready</span> <span m="2653250">to</span> <span m="2653360">state</span> <span m="2653780">the</span> <span m="2653910">invariant.</span> <span m="2664310">Anybody</span> <span m="2664600">tell</span> <span m="2664800">me</span> <span m="2664910">what</span> <span m="2665070">the</span> <span m="2665190">invariant</span> <span m="2667170">is</span> <span m="2667300">going</span> <span m="2667410">to</span> <span m="2667470">be?</span> <span m="2667930">What's</span> <span m="2668120">the</span> <span m="2668220">invariant</span> <span m="2668660">going</span> <span m="2668770">to</span> <span m="2668810">be</span> <span m="2668950">in</span> <span m="2669040">this</span> <span m="2669130">system?</span> </p>
<p><span m="2671805">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="2673140">PROFESSOR: Yeah,</span> <span m="2673840">the</span> <span m="2674670">parity</span> <span m="2675360">of</span> <span m="2675510">the</span> <span m="2675610">number</span> <span m="2675950">of</span> <span m="2676020">inversions</span> <span m="2677630">is</span> <span m="2677900">odd,</span> <span m="2679900">because</span> <span m="2680120">it</span> <span m="2680240">starts</span> <span m="2680630">that</span> <span m="2680820">way.</span> <span m="2681650">It's</span> <span m="2681860">preserved.</span> <span m="2682960">And</span> <span m="2683110">it</span> <span m="2683200">won't</span> <span m="2683520">hold</span> <span m="2684290">in</span> <span m="2684630">the</span> <span m="2684770">desired</span> <span m="2685230">end</span> <span m="2685400">state</span> <span m="2685720">that</span> <span m="2685780">we're</span> <span m="2685870">looking</span> <span m="2686130">at.</span> </p>
<p><span m="2687360">So</span> <span m="2687480">we</span> <span m="2687560">can</span> <span m="2687670">state</span> <span m="2687940">that</span> <span m="2688100">as</span> <span m="2688220">follows.</span> <span m="2695240">In</span> <span m="2695570">every</span> <span m="2697350">state</span> <span m="2697890">or</span> <span m="2698110">configuration</span> <span m="2699950">reachable</span> <span m="2705810">from</span> <span m="2706830">the</span> <span m="2706970">start</span> <span m="2707300">state,</span> <span m="2712380">which</span> <span m="2712510">is</span> <span m="2712820">a,</span> <span m="2713530">b,</span> <span m="2713972">c,</span> <span m="2714414">d,</span> <span m="2714856">e,</span> <span m="2715740">f,</span> <span m="2716120">h,</span> <span m="2716500">g--</span> <span m="2717260">which</span> <span m="2717480">are</span> <span m="2717560">out</span> <span m="2717680">of</span> <span m="2717770">order--</span> <span m="2721160">the</span> <span m="2721300">parity</span> <span m="2724255">of</span> <span m="2724710">the</span> <span m="2724910">number</span> <span m="2725790">of</span> <span m="2726030">inversions</span> <span m="2729990">is</span> <span m="2730250">odd.</span> <span m="2735220">And</span> <span m="2735330">the</span> <span m="2735400">proof</span> <span m="2735910">will</span> <span m="2736010">be</span> <span m="2736350">by</span> <span m="2736560">induction.</span> <span m="2738440">And</span> <span m="2738580">invariant</span> <span m="2739140">proofs</span> <span m="2739480">are</span> <span m="2739570">always</span> <span m="2740070">by</span> <span m="2740260">induction.</span> </p>
<p><span m="2746790">And</span> <span m="2747100">the</span> <span m="2747180">inductive</span> <span m="2747660">hypothesis--</span> <span m="2749780">and</span> <span m="2749960">this</span> <span m="2750100">is</span> <span m="2750210">very</span> <span m="2750570">typical</span> <span m="2751180">in a proof</span> <span m="2751550">by</span> <span m="2752000">using</span> <span m="2752270">invariants--</span> <span m="2752870">is,</span> <span m="2753820">so</span> <span m="2753950">P</span> <span m="2754170">of</span> <span m="2754240">n</span> <span m="2755990">is</span> <span m="2757040">after</span> <span m="2759710">any</span> <span m="2760630">sequence</span> <span m="2765160">of n</span> <span m="2765450">moves</span> <span m="2771910">from</span> <span m="2772370">the</span> <span m="2772460">start</span> <span m="2772770">state--</span> <span m="2774880">in</span> <span m="2774940">fact,</span> <span m="2776320">just</span> <span m="2776610">the</span> <span m="2776750">rest</span> <span m="2776970">of</span> <span m="2777030">this</span> <span m="2777260">is</span> <span m="2777340">what</span> <span m="2777490">it</span> <span m="2777540">is.</span> <span m="2785700">All</span> <span m="2785820">right,</span> <span m="2786020">so</span> <span m="2786200">our</span> <span m="2786650">inductive</span> <span m="2786710">hypothesis</span> <span m="2787480">is</span> <span m="2788690">P</span> <span m="2788900">of</span> <span m="2789000">n,</span> <span m="2789280">after</span> <span m="2789630">any</span> <span m="2789980">sequence</span> <span m="2790500">of</span> <span m="2790730">n</span> <span m="2791050">moves--</span> <span m="2791740">so</span> <span m="2792020">n</span> <span m="2792730">is</span> <span m="2792880">the</span> <span m="2792950">number</span> <span m="2793170">of</span> <span m="2793230">moves</span> <span m="2793510">you</span> <span m="2793610">took</span> <span m="2794270">to</span> <span m="2794360">get</span> <span m="2794580">there--</span> <span m="2795460">from</span> <span m="2795870">the</span> <span m="2795970">start</span> <span m="2796280">state,</span> <span m="2797510">the</span> <span m="2797640">parity</span> <span m="2799200">of</span> <span m="2799320">the</span> <span m="2799410">number</span> <span m="2799590">of</span> <span m="2799630">inversions</span> <span m="2800100">is</span> <span m="2800230">odd.</span> <span m="2801550">That</span> <span m="2801960">is</span> <span m="2802170">the</span> <span m="2802260">inductive</span> <span m="2802610">hypothesis.</span> <span m="2804180">This</span> <span m="2804440">is</span> <span m="2804570">the</span> <span m="2804720">invariant.</span> </p>
<p><span m="2805680">And</span> <span m="2805810">they</span> <span m="2805910">become</span> <span m="2806420">one</span> <span m="2806570">and</span> <span m="2806640">the</span> <span m="2806700">same.</span> <span m="2807380">And</span> <span m="2807550">the</span> <span m="2807630">parameter</span> <span m="2808110">n</span> <span m="2808780">is</span> <span m="2808930">the</span> <span m="2809010">number</span> <span m="2809410">of</span> <span m="2809490">moves</span> <span m="2809850">you've</span> <span m="2809950">taken.</span> <span m="2810960">And</span> <span m="2811080">that's</span> <span m="2811300">how</span> <span m="2811400">you</span> <span m="2811550">set</span> <span m="2811800">up</span> <span m="2812420">a</span> <span m="2812520">proof</span> <span m="2812920">using an</span> <span m="2813240">invariant.</span> <span m="2813625">It</span> <span m="2814010">always</span> <span m="2814450">looks</span> <span m="2814850">just</span> <span m="2815110">like</span> <span m="2815280">this.</span> </p>
<p><span m="2836550">OK,</span> <span m="2836810">so</span> <span m="2836950">we're</span> <span m="2837020">doing</span> <span m="2837230">a</span> <span m="2837290">proof</span> <span m="2837570">by</span> <span m="2837750">induction.</span> <span m="2839220">We've</span> <span m="2839450">got</span> <span m="2839790">the</span> <span m="2841110">inductive</span> <span m="2841520">hypothesis.</span> <span m="2842720">What's</span> <span m="2843030">the</span> <span m="2843110">next</span> <span m="2843330">step?</span> </p>
<p><span m="2845235">AUDIENCE: Base case.</span> </p>
<p><span m="2845670">PROFESSOR: Base</span> <span m="2846010">case.</span> <span m="2852590">And</span> <span m="2852700">in</span> <span m="2852780">these</span> <span m="2853000">circumstances,</span> <span m="2853770">where</span> <span m="2853860">the</span> <span m="2853970">proof</span> <span m="2854250">by</span> <span m="2854440">invariant,</span> <span m="2854830">the</span> <span m="2854930">base</span> <span m="2855210">case</span> <span m="2855480">is</span> <span m="2855610">always</span> <span m="2855930">0.</span> <span m="2857590">We</span> <span m="2857700">haven't</span> <span m="2857960">made</span> <span m="2858170">any</span> <span m="2858410">moves</span> <span m="2858680">yet.</span> <span m="2860020">Now</span> <span m="2860170">in</span> <span m="2860230">this</span> <span m="2860420">case,</span> <span m="2860780">if</span> <span m="2860790">we</span> <span m="2860870">haven't</span> <span m="2861090">made</span> <span m="2861390">moves,</span> <span m="2861860">the</span> <span m="2861920">number</span> <span m="2862160">of</span> <span m="2862210">inversions</span> <span m="2864410">in</span> <span m="2864510">the</span> <span m="2864580">start</span> <span m="2864860">state</span> <span m="2865170">now</span> <span m="2866700">is</span> <span m="2866880">1.</span> </p>
<p><span m="2868870">That</span> <span m="2869060">means</span> <span m="2869310">the</span> <span m="2869410">parity</span> <span m="2869950">is</span> <span m="2870080">odd,</span> <span m="2873250">and</span> <span m="2873700">the</span> <span m="2874650">hypothesis</span> <span m="2875380">is</span> <span m="2875470">satisfied.</span> <span m="2876230">P</span> <span m="2876430">of</span> <span m="2876490">0</span> <span m="2876890">is</span> <span m="2876990">true,</span> <span m="2877870">because</span> <span m="2878230">in</span> <span m="2878320">any</span> <span m="2878510">sequence</span> <span m="2878960">of</span> <span m="2879060">zero</span> <span m="2879570">moves,</span> <span m="2880650">after</span> <span m="2880990">this--</span> <span m="2881640">i.e.</span> <span m="2882250">in</span> <span m="2882370">this</span> <span m="2882550">state--</span> <span m="2882860">you've</span> <span m="2882960">got</span> <span m="2883140">one</span> <span m="2883460">inversion.</span> <span m="2883800">That's</span> <span m="2884010">odd.</span> </p>
<p><span m="2886602">The</span> <span m="2887040">last</span> <span m="2887350">step</span> <span m="2887630">is</span> <span m="2887860">the</span> <span m="2888080">inductive</span> <span m="2888530">step.</span> <span m="2896690">And</span> <span m="2896810">here</span> <span m="2897110">we</span> <span m="2897180">need</span> <span m="2897330">to</span> <span m="2897390">show</span> <span m="2898480">for</span> <span m="2898980">any</span> <span m="2899330">n</span> <span m="2899990">bigger</span> <span m="2900250">or</span> <span m="2900320">equal</span> <span m="2900500">to</span> <span m="2900570">0,</span> <span m="2902510">we</span> <span m="2902630">need</span> <span m="2902810">to</span> <span m="2902880">show</span> <span m="2904970">P</span> <span m="2905280">of</span> <span m="2905370">n</span> <span m="2906140">implies</span> <span m="2906880">p</span> <span m="2907100">of</span> <span m="2907180">n</span> <span m="2907300">plus</span> <span m="2907580">1</span> <span m="2907820">is</span> <span m="2907990">true--</span> <span m="2909980">the</span> <span m="2910060">standard</span> <span m="2910540">thing.</span> <span m="2912990">So</span> <span m="2913180">let's</span> <span m="2913440">look</span> <span m="2913580">at</span> <span m="2913690">P n</span> <span m="2914120">plus</span> <span m="2914420">1.</span> <span m="2915770">That's</span> <span m="2916020">talking</span> <span m="2916400">about</span> <span m="2917380">where</span> <span m="2917660">we</span> <span m="2917850">are</span> <span m="2918260">after</span> <span m="2918640">a</span> <span m="2918690">sequence</span> <span m="2919210">of</span> <span m="2919400">n</span> <span m="2919780">plus</span> <span m="2920180">1</span> <span m="2920530">moves</span> <span m="2922090">in</span> <span m="2922200">the</span> <span m="2922290">puzzle,</span> <span m="2922550">from</span> <span m="2922830">the</span> <span m="2922970">start</span> <span m="2923250">state.</span> </p>
<p><span m="2925610">So</span> <span m="2925800">let's</span> <span m="2926060">consider</span> <span m="2930450">any</span> <span m="2930770">sequence</span> <span m="2934865">of</span> <span m="2935360">n</span> <span m="2935610">plus</span> <span m="2935890">1</span> <span m="2936110">moves.</span> <span m="2938790">And</span> <span m="2938950">call</span> <span m="2939310">these</span> <span m="2939550">moves--</span> <span m="2940080">label</span> <span m="2940430">them by</span> <span m="2940660">M1</span> <span m="2942470">all</span> <span m="2942730">the</span> <span m="2942810">way</span> <span m="2942940">to</span> <span m="2943040">M</span> <span m="2943720">n</span> <span m="2944060">plus</span> <span m="2944320">1.</span> <span m="2945520">Those</span> <span m="2945790">are</span> <span m="2945840">the</span> <span m="2945910">moves</span> <span m="2946180">we</span> <span m="2946270">made.</span> </p>
<p><span m="2949080">Now,</span> <span m="2949470">by</span> <span m="2949740">the</span> <span m="2949870">inductive</span> <span m="2950290">hypothesis,</span> <span m="2951100">because</span> <span m="2951670">of</span> <span m="2951790">P n,</span> <span m="2953680">we</span> <span m="2953820">know</span> <span m="2954060">that</span> <span m="2954190">after</span> <span m="2954400">the</span> <span m="2954490">first</span> <span m="2954910">n</span> <span m="2955280">moves,</span> <span m="2956170">the</span> <span m="2956260">parity</span> <span m="2956640">is still</span> <span m="2956880">odd.</span> <span m="2957720">That's</span> <span m="2957960">what</span> <span m="2958110">P</span> <span m="2958290">n</span> <span m="2958490">says.</span> <span m="2960170">That's</span> <span m="2960370">P</span> <span m="2960675">n.</span> </p>
<p><span m="2960980">After</span> <span m="2961330">any</span> <span m="2961580">sequence</span> <span m="2962080">of</span> <span m="2962260">n</span> <span m="2962520">moves</span> <span m="2963650">from</span> <span m="2963860">here,</span> <span m="2964810">the</span> <span m="2964930">parity</span> <span m="2965420">of the</span> <span m="2965510">number</span> <span m="2965710">of</span> <span m="2965800">inversions</span> <span m="2966240">is odd.</span> <span m="2969490">So</span> <span m="2969650">we</span> <span m="2969710">could</span> <span m="2969830">say</span> <span m="2969990">by</span> <span m="2970170">the</span> <span m="2970250">inductive</span> <span m="2970630">hypothesis,</span> <span m="2971460">or</span> <span m="2971500">we</span> <span m="2971580">could</span> <span m="2971730">say</span> <span m="2972160">by</span> <span m="2972390">P</span> <span m="2972730">n,</span> <span m="2975230">we</span> <span m="2975380">know</span> <span m="2980220">that</span> <span m="2981930">the</span> <span m="2982040">parity</span> <span m="2986220">after</span> <span m="2986690">moves</span> <span m="2988450">M1</span> <span m="2990030">out</span> <span m="2990320">to</span> <span m="2990510">M n</span> <span m="2993820">is</span> <span m="2994020">odd.</span> <span m="3000130">Now,</span> <span m="3001050">by</span> <span m="3001210">Corollary</span> <span m="3001890">One--</span> <span m="3002330">do</span> <span m="3002400">we</span> <span m="3002480">still</span> <span m="3002660">have</span> <span m="3002790">Corollary</span> <span m="3003250">One?</span> <span m="3005240">[INAUDIBLE]</span> </p>
<p><span m="3007205">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="3007640">PROFESSOR: [INAUDIBLE],</span> <span m="3008040">yeah,</span> <span m="3008190">good,</span> <span m="3008530">we've</span> <span m="3008620">got</span> <span m="3008760">Corollary</span> <span m="3009130">One.</span> <span m="3009980">During</span> <span m="3010650">any</span> <span m="3010910">move,</span> <span m="3011320">the</span> <span m="3011440">parity</span> <span m="3012700">doesn't</span> <span m="3013040">change.</span> <span m="3014650">That</span> <span m="3014830">says</span> <span m="3015210">from</span> <span m="3017320">after</span> <span m="3017550">the</span> <span m="3017690">nth</span> <span m="3017900">move</span> <span m="3018160">to</span> <span m="3018350">after</span> <span m="3018450">the</span> <span m="3018540">n</span> <span m="3018640">plus</span> <span m="3018880">1st</span> <span m="3019190">move--</span> <span m="3019630">one</span> <span m="3019870">extra</span> <span m="3020120">move--</span> <span m="3020700">doesn't</span> <span m="3021020">change</span> <span m="3021340">the</span> <span m="3021430">parity.</span> <span m="3022330">So</span> <span m="3022590">it's</span> <span m="3022700">still</span> <span m="3022870">odd.</span> </p>
<p></p>
<p><span m="3040600">So</span> <span m="3040700">next,</span> <span m="3040900">by</span> <span m="3041060">Corollary</span> <span m="3041670">One,</span> <span m="3046820">we</span> <span m="3047000">know</span> <span m="3050080">that</span> <span m="3050210">the</span> <span m="3050290">parity</span> <span m="3052810">of</span> <span m="3053230">the</span> <span m="3053410">number</span> <span m="3053740">of</span> <span m="3053790">inversions</span> <span m="3059010">does</span> <span m="3059180">not</span> <span m="3059460">change</span> <span m="3060530">during</span> <span m="3061100">M n</span> <span m="3061430">plus</span> <span m="3061640">1.</span> <span m="3074744">Therefore,</span> <span m="3076670">this</span> <span m="3076870">implies</span> <span m="3077430">that</span> <span m="3077590">the</span> <span m="3077680">parity</span> <span m="3083780">after</span> <span m="3084130">all</span> <span m="3085110">n</span> <span m="3085300">plus</span> <span m="3085500">1</span> <span m="3085710">moves--</span> <span m="3086840">these</span> <span m="3087050">guys--</span> <span m="3094360">is</span> <span m="3094620">odd.</span> <span m="3095340">It</span> <span m="3095750">started</span> <span m="3096210">odd.</span> </p>
<p><span m="3097180">It</span> <span m="3097310">stayed</span> <span m="3097690">odd</span> <span m="3097900">after</span> <span m="3098100">the</span> <span m="3098190">first</span> <span m="3098450">n.</span> <span m="3099150">It</span> <span m="3099290">stayed</span> <span m="3099710">odd</span> <span m="3100040">during</span> <span m="3100240">the</span> <span m="3100340">n</span> <span m="3100470">plus</span> <span m="3100750">1st.</span> <span m="3101580">So</span> <span m="3101900">it's</span> <span m="3102070">odd.</span> <span m="3103940">And</span> <span m="3103990">that's</span> <span m="3104180">exactly</span> <span m="3105720">the</span> <span m="3105810">statement</span> <span m="3106220">of</span> <span m="3106320">P n</span> <span m="3106650">plus</span> <span m="3106900">1.</span> </p>
<p><span m="3108780">So</span> <span m="3109220">we've</span> <span m="3109400">completed</span> <span m="3109820">the</span> <span m="3109910">induction.</span> <span m="3110740">We've</span> <span m="3110950">shown</span> <span m="3111330">that</span> <span m="3111440">P n</span> <span m="3111910">implies</span> <span m="3112300">P n</span> <span m="3112540">plus</span> <span m="3112790">1.</span> <span m="3116450">All</span> <span m="3116630">right,</span> <span m="3116850">so</span> <span m="3116940">now,</span> <span m="3118640">any</span> <span m="3118790">questions</span> <span m="3119210">so</span> <span m="3119340">far</span> <span m="3120110">on</span> <span m="3120480">this?</span> <span m="3121241">Yeah.</span> </p>
<p><span m="3122143">AUDIENCE: What's the difference</span> <span m="3122594">between</span> <span m="3123045">a</span> <span m="3123496">lemma</span> <span m="3123947">and a corollary?</span> </p>
<p><span m="3125300">PROFESSOR: Really,</span> <span m="3126440">there's</span> <span m="3126610">not</span> <span m="3126720">a</span> <span m="3126740">lot</span> <span m="3126980">of</span> <span m="3127030">difference.</span> <span m="3127440">Corollary</span> <span m="3128120">usually</span> <span m="3128810">is</span> <span m="3128910">something</span> <span m="3129280">that's</span> <span m="3129450">a</span> <span m="3129510">simple</span> <span m="3130050">consequence</span> <span m="3131430">of</span> <span m="3131550">something</span> <span m="3131890">else.</span> <span m="3132870">And</span> <span m="3134130">the</span> <span m="3134240">corollary</span> <span m="3135490">had</span> <span m="3135610">a</span> <span m="3135670">pretty</span> <span m="3135950">short</span> <span m="3136340">proof.</span> <span m="3137000">Usually</span> <span m="3137430">when</span> <span m="3137500">you</span> <span m="3137560">see</span> <span m="3137720">corollary,</span> <span m="3138180">that's</span> <span m="3138270">not</span> <span m="3138410">a</span> <span m="3138450">10-page</span> <span m="3138900">proof</span> <span m="3139150">there.</span> </p>
<p><span m="3139770">A</span> <span m="3140150">lemma</span> <span m="3141390">is</span> <span m="3141790">something that</span> <span m="3142180">you're</span> <span m="3142300">going</span> <span m="3142410">to</span> <span m="3142450">use</span> <span m="3142900">as</span> <span m="3143010">a</span> <span m="3143100">tool</span> <span m="3143480">for</span> <span m="3143650">a</span> <span m="3143700">bigger</span> <span m="3144020">thing,</span> <span m="3144710">often</span> <span m="3145010">a</span> <span m="3145090">theorem</span> <span m="3145500">or</span> <span m="3145630">another</span> <span m="3145860">lemma.</span> <span m="3147060">In fact,</span> <span m="3147290">now</span> <span m="3147530">we're</span> <span m="3147640">going</span> <span m="3147740">to</span> <span m="3147780">get</span> <span m="3148040">to</span> <span m="3148880">the</span> <span m="3149010">theorem,</span> <span m="3149720">which</span> <span m="3149970">is</span> <span m="3150110">sort</span> <span m="3150330">of</span> <span m="3150390">the</span> <span m="3150480">final</span> <span m="3150990">thing--</span> <span m="3151180">the</span> <span m="3151270">thing</span> <span m="3151450">we're</span> <span m="3151640">really</span> <span m="3152000">after.</span> <span m="3152820">And</span> <span m="3152870">that's</span> <span m="3153020">sort</span> <span m="3153180">of</span> <span m="3153250">the</span> <span m="3153290">big</span> <span m="3153540">deal.</span> <span m="3154540">And</span> <span m="3154710">now</span> <span m="3154840">the</span> <span m="3154940">proof</span> <span m="3155150">will</span> <span m="3155210">be</span> <span m="3155320">simple,</span> <span m="3155690">because</span> <span m="3156060">we've</span> <span m="3156210">done</span> <span m="3156400">all</span> <span m="3156530">these</span> <span m="3156710">lemmas</span> <span m="3157000">and</span> <span m="3157090">corollaries.</span> <span m="3158800">Any</span> <span m="3158950">other</span> <span m="3159080">questions?</span> <span m="3162230">All right.</span> </p>
<p><span m="3164840">In</span> <span m="3164880">fact,</span> <span m="3165900">the</span> <span m="3166010">theorem</span> <span m="3167050">is</span> <span m="3167450">up</span> <span m="3167590">here,</span> <span m="3167850">right?</span> <span m="3169170">There</span> <span m="3169330">is</span> <span m="3169510">no</span> <span m="3169760">sequence</span> <span m="3170200">of</span> <span m="3170300">legal</span> <span m="3170530">moves</span> <span m="3170820">to</span> <span m="3170940">invert</span> <span m="3171280">g and h.</span> <span m="3172040">In</span> <span m="3172080">other</span> <span m="3172250">words,</span> <span m="3172890">the</span> <span m="3172900">students</span> <span m="3173280">were</span> <span m="3173350">screwed.</span> <span m="3173950">They</span> <span m="3174060">couldn't</span> <span m="3174360">do</span> <span m="3174550">it</span> <span m="3175890">fairly.</span> <span m="3177060">So</span> <span m="3177180">let's</span> <span m="3177350">prove</span> <span m="3177670">the</span> <span m="3177760">theorem</span> <span m="3178120">now.</span> </p>
<p></p>
<p><span m="3208590">The</span> <span m="3208720">parity</span> <span m="3211795">of</span> <span m="3212250">the</span> <span m="3212420">number</span> <span m="3212770">of</span> <span m="3212850">inversions</span> <span m="3219030">in</span> <span m="3219320">the</span> <span m="3220900">desired</span> <span m="3221460">state--</span> <span m="3221930">i.e.</span> <span m="3222520">in</span> <span m="3222740">order--</span> <span m="3224520">the</span> <span m="3224630">target</span> <span m="3225050">state--</span> <span m="3229260">is</span> <span m="3229480">even.</span> <span m="3231050">Nothing's</span> <span m="3231440">out</span> <span m="3231560">of</span> <span m="3231660">order.</span> <span m="3232160">There's</span> <span m="3232350">zero</span> <span m="3233040">inversions.</span> </p>
<p><span m="3235320">By</span> <span m="3235480">Lemma</span> <span m="3235750">Four,</span> <span m="3242300">the</span> <span m="3242400">desired</span> <span m="3242960">state</span> <span m="3245530">cannot</span> <span m="3246060">be</span> <span m="3246200">reached</span> <span m="3246990">from</span> <span m="3247030">the</span> <span m="3247120">start</span> <span m="3247400">state,</span> <span m="3249840">because</span> <span m="3250010">its</span> <span m="3250140">parity</span> <span m="3250450">is</span> <span m="3250580">odd,</span> <span m="3266450">using</span> <span m="3266830">legal</span> <span m="3267070">moves.</span> <span m="3267750">So</span> <span m="3267910">we're</span> <span m="3268000">done.</span> <span m="3268420">So</span> <span m="3268620">the proof of</span> <span m="3268880">the</span> <span m="3268980">theorem--</span> <span m="3269890">by</span> <span m="3270010">now,</span> <span m="3270290">it's</span> <span m="3270670">short,</span> <span m="3271600">because</span> <span m="3271770">we</span> <span m="3271870">did</span> <span m="3272040">four</span> <span m="3272300">lemmas,</span> <span m="3272690">a</span> <span m="3272740">corollary,</span> <span m="3273410">and</span> <span m="3273960">lots</span> <span m="3274200">of</span> <span m="3274300">argument.</span> </p>
<p><span m="3278850">Could</span> <span m="3279210">you</span> <span m="3279420">have</span> <span m="3279560">solved</span> <span m="3280010">the</span> <span m="3280100">puzzle</span> <span m="3281330">if</span> <span m="3281510">I</span> <span m="3281630">allow</span> <span m="3282100">the</span> <span m="3282200">blank</span> <span m="3282790">to</span> <span m="3282870">be</span> <span m="3283010">somewhere</span> <span m="3283420">else</span> <span m="3283640">at</span> <span m="3283710">the</span> <span m="3283870">end?</span> <span m="3286680">Is</span> <span m="3286900">the</span> <span m="3287000">puzzle</span> <span m="3287390">solvable</span> <span m="3287840">then?</span> <span m="3291180">No.</span> <span m="3292250">Why</span> <span m="3292430">not?</span> <span m="3295770">Doesn't</span> <span m="3296170">impact</span> <span m="3296610">the</span> <span m="3296690">parity at all,</span> <span m="3297040">so</span> <span m="3297390">you</span> <span m="3297480">can't</span> <span m="3297720">do</span> <span m="3297830">that.</span> </p>
<p><span m="3304450">Again,</span> <span m="3304870">the</span> <span m="3305200">idea</span> <span m="3305630">here--</span> <span m="3306130">and</span> <span m="3306210">you'll</span> <span m="3306300">get</span> <span m="3306420">some</span> <span m="3306560">practice</span> <span m="3306930">in</span> <span m="3307000">homework--</span> <span m="3308310">is</span> <span m="3308850">you're</span> <span m="3309000">looking</span> <span m="3309410">for</span> <span m="3309540">a</span> <span m="3309600">property</span> <span m="3310570">that</span> <span m="3310690">holds at</span> <span m="3311070">the</span> <span m="3311150">beginning,</span> <span m="3312140">is</span> <span m="3312320">preserved</span> <span m="3312820">by</span> <span m="3312970">every</span> <span m="3313270">step,</span> <span m="3314500">but</span> <span m="3314700">is</span> <span m="3314870">not</span> <span m="3315400">present</span> <span m="3316290">in</span> <span m="3316450">the</span> <span m="3316560">target</span> <span m="3316940">state.</span> <span m="3319190">Now</span> <span m="3319960">actually,</span> <span m="3320270">this</span> <span m="3320470">puzzle</span> <span m="3320860">was</span> <span m="3321010">enormously</span> <span m="3321710">popular</span> <span m="3322980">in</span> <span m="3323160">the</span> <span m="3323770">late</span> <span m="3324150">1880s.</span> <span m="3325260">In</span> <span m="3325370">fact</span> <span m="3325750">there</span> <span m="3325880">was</span> <span m="3326000">$1,000</span> <span m="3327110">prize</span> <span m="3327630">offered</span> <span m="3328710">for</span> <span m="3328860">anybody</span> <span m="3329260">who</span> <span m="3329380">could</span> <span m="3329520">solve</span> <span m="3330070">the</span> <span m="3330880">larger</span> <span m="3331290">version.</span> </p>
<p><span m="3331640">Back</span> <span m="3331890">then,</span> <span m="3332050">it</span> <span m="3332100">was</span> <span m="3332400">the</span> <span m="3332480">15</span> <span m="3333010">puzzle,</span> <span m="3333720">which</span> <span m="3333880">you</span> <span m="3333960">can</span> <span m="3334080">still</span> <span m="3334310">buy</span> <span m="3334550">today--</span> <span m="3335360">same</span> <span m="3335700">problem</span> <span m="3336150">on</span> <span m="3336240">a</span> <span m="3336290">4</span> <span m="3336560">by</span> <span m="3336700">4</span> <span m="3337020">grid--</span> <span m="3337740">exactly</span> <span m="3338170">the</span> <span m="3338260">same</span> <span m="3338460">problem.</span> <span m="3339150">A</span> <span m="3339240">pair</span> <span m="3339550">is</span> <span m="3339650">out</span> <span m="3339780">of</span> <span m="3339870">order.</span> <span m="3340470">You've</span> <span m="3340570">got</span> <span m="3340740">to</span> <span m="3340800">get</span> <span m="3340950">them</span> <span m="3341190">in</span> <span m="3341390">order.</span> <span m="3342430">You</span> <span m="3342550">can't</span> <span m="3342830">do</span> <span m="3343010">it.</span> </p>
<p><span m="3344510">Now,</span> <span m="3344670">the</span> <span m="3344810">proof</span> <span m="3345070">you</span> <span m="3345180">can't</span> <span m="3345450">do</span> <span m="3345630">it</span> <span m="3346270">is</span> <span m="3346460">a</span> <span m="3346510">little</span> <span m="3346750">harder--</span> <span m="3347400">same</span> <span m="3347960">idea--</span> <span m="3348610">one</span> <span m="3349040">extra</span> <span m="3349400">idea</span> <span m="3350210">in</span> <span m="3350390">it--</span> <span m="3351630">one</span> <span m="3351810">extra</span> <span m="3352050">lemma.</span> <span m="3352570">And</span> <span m="3352730">that's</span> <span m="3352940">for</span> <span m="3353050">homework.</span> <span m="3354080">So</span> <span m="3354300">you'll</span> <span m="3354590">go</span> <span m="3354790">through</span> <span m="3355060">and</span> <span m="3355520">see</span> <span m="3355680">exactly</span> <span m="3356160">the</span> <span m="3356280">structure</span> <span m="3356700">with</span> <span m="3356820">one</span> <span m="3357000">extra</span> <span m="3357260">little</span> <span m="3357420">trick</span> <span m="3357670">it.</span> </p>
<p><span m="3359360">But</span> <span m="3360370">back</span> <span m="3360620">then,</span> <span m="3360810">I</span> <span m="3360870">guess</span> <span m="3361100">most</span> <span m="3361320">people</span> <span m="3361530">didn't--</span> <span m="3362350">of</span> <span m="3362410">course,</span> <span m="3362670">most people</span> <span m="3362870">didn't</span> <span m="3363130">realize</span> <span m="3363420">it</span> <span m="3363470">wasn't</span> <span m="3363710">doable.</span> <span m="3364010">So</span> <span m="3364110">they</span> <span m="3364180">could</span> <span m="3364290">offer</span> <span m="3364580">this</span> <span m="3364750">price</span> <span m="3365150">safely.</span> <span m="3366250">$1,000</span> <span m="3366920">then is probably</span> <span m="3367410">worth</span> <span m="3367580">a</span> <span m="3367630">quarter</span> <span m="3367870">million</span> <span m="3368120">dollars</span> <span m="3368540">today.</span> <span m="3371130">Any</span> <span m="3371290">questions</span> <span m="3371750">on</span> <span m="3371950">the</span> <span m="3372510">eight</span> <span m="3372710">puzzle?</span> <span m="3373440">Yeah.</span> </p>
<p><span m="3375624">AUDIENCE: The</span> <span m="3376120">inductive</span> <span m="3376616">steps--</span> <span m="3379096">when you said,</span> <span m="3379592">"by inductive</span> <span m="3380088">hypothesis,"</span> <span m="3382072">[INAUDIBLE]</span> <span m="3382568">parity</span> <span m="3383064">after,</span> <span m="3383560">shouldn't</span> <span m="3384056">the first</span> <span m="3384552">term be</span> <span m="3385048">M0?</span> </p>
<p><span m="3388030">PROFESSOR: Let's</span> <span m="3388220">see.</span> <span m="3388340">No,</span> <span m="3390190">the</span> <span m="3390260">first</span> <span m="3390640">move</span> <span m="3390810">is</span> <span m="3390920">M1.</span> <span m="3391250">The</span> <span m="3391580">second</span> <span m="3391920">is</span> <span m="3392230">M2.</span> <span m="3393410">Now,</span> <span m="3393600">when I</span> <span m="3393730">talk</span> <span m="3394030">about</span> <span m="3394780">zero</span> <span m="3395240">moves,</span> <span m="3397140">that</span> <span m="3397320">means</span> <span m="3397520">no</span> <span m="3397630">moves are</span> <span m="3397880">taking</span> <span m="3398100">place.</span> <span m="3398400">I'm</span> <span m="3398480">still</span> <span m="3398650">in</span> <span m="3398740">the</span> <span m="3398810">start</span> <span m="3399110">state.</span> </p>
<p><span m="3399970">So</span> <span m="3400190">a move</span> <span m="3400470">is</span> <span m="3400570">a</span> <span m="3400640">transition</span> <span m="3401260">between</span> <span m="3401630">states.</span> <span m="3402920">So</span> <span m="3402930">you</span> <span m="3402990">could</span> <span m="3403120">have</span> <span m="3403290">state</span> <span m="3403390">0.</span> <span m="3404160">That's</span> <span m="3404380">the</span> <span m="3404440">start</span> <span m="3404730">state.</span> <span m="3405556">State</span> <span m="3405970">one</span> <span m="3406210">would</span> <span m="3406310">be</span> <span m="3406410">the</span> <span m="3406490">move</span> <span m="3407400">after</span> <span m="3407660">move</span> <span m="3407840">one.</span> </p>
<p><span m="3409980">Have I</span> <span m="3410060">got</span> <span m="3410220">that</span> <span m="3410380">right</span> <span m="3410580">here?</span> <span m="3410810">So</span> <span m="3411290">the</span> <span m="3411560">inductive</span> <span m="3412020">step,</span> <span m="3414720">I</span> <span m="3414790">need</span> <span m="3415000">to</span> <span m="3415060">show</span> <span m="3415270">P0</span> <span m="3415450">zero</span> <span m="3415700">implies</span> <span m="3416040">P1.</span> <span m="3416550">P1</span> <span m="3416640">implies</span> <span m="3417100">P2,</span> <span m="3417470">and</span> <span m="3417570">so</span> <span m="3417750">forth.</span> <span m="3419070">And</span> <span m="3419280">P n</span> <span m="3419950">is</span> <span m="3420540">the</span> <span m="3420630">parity</span> <span m="3420990">of</span> <span m="3421230">inversions</span> <span m="3421960">after</span> <span m="3422900">n</span> <span m="3423330">moves.</span> </p>
<p><span m="3426200">So</span> <span m="3426340">you</span> <span m="3426470">could</span> <span m="3426650">have</span> <span m="3426820">state</span> <span m="3427220">s0.</span> <span m="3427670">We</span> <span m="3427770">didn't</span> <span m="3428560">label</span> <span m="3428850">the</span> <span m="3428940">states</span> <span m="3429290">here,</span> <span m="3429360">but you</span> <span m="3429560">could</span> <span m="3429700">have</span> <span m="3429800">state</span> <span m="3430250">s0.</span> <span m="3432835">Yeah.</span> </p>
<p><span m="3433817">AUDIENCE: So in this</span> <span m="3434308">problem,</span> <span m="3435290">we</span> <span m="3435781">assume</span> <span m="3436272">that</span> <span m="3436763">everything</span> <span m="3437254">should be</span> <span m="3438236">defined</span> <span m="3438727">as</span> <span m="3439220">rows</span> <span m="3439560">or columns.</span> <span m="3440830">How</span> <span m="3441329">do we</span> <span m="3441828">know</span> <span m="3442327">when</span> <span m="3442826">the</span> <span m="3443325">assumptions</span> <span m="3443824">are</span> <span m="3444323">valid?</span> </p>
<p><span m="3446320">PROFESSOR: That's</span> <span m="3446490">a good</span> <span m="3446560">question.</span> <span m="3446870">OK,</span> <span m="3447080">so</span> <span m="3447250">really,</span> <span m="3447480">there's</span> <span m="3448510">two</span> <span m="3448790">cases.</span> <span m="3451470">To</span> <span m="3451540">be</span> <span m="3451670">really</span> <span m="3451980">precise,</span> <span m="3452240">we</span> <span m="3452320">should</span> <span m="3452490">have</span> <span m="3452570">argued</span> <span m="3452950">there's</span> <span m="3453090">only</span> <span m="3453250">two</span> <span m="3453380">types</span> <span m="3453640">of</span> <span m="3453710">moves,</span> <span m="3454570">a</span> <span m="3454680">row</span> <span m="3454880">move and</span> <span m="3455100">a</span> <span m="3455170">column</span> <span m="3455510">move.</span> </p>
<p><span m="3456100">And</span> <span m="3456300">in</span> <span m="3456370">fact,</span> <span m="3456800">if</span> <span m="3456900">you</span> <span m="3456920">look</span> <span m="3457140">at</span> <span m="3457240">what</span> <span m="3457450">can move</span> <span m="3457750">into</span> <span m="3457920">a</span> <span m="3457970">blank</span> <span m="3458250">square,</span> <span m="3458620">there's</span> <span m="3458850">four</span> <span m="3459250">guys</span> <span m="3459600">that</span> <span m="3459680">can</span> <span m="3459860">do</span> <span m="3460050">it--</span> <span m="3460470">the</span> <span m="3460560">guy</span> <span m="3460730">above,</span> <span m="3461100">the</span> <span m="3461180">guy</span> <span m="3461330">to the</span> <span m="3461470">right,</span> <span m="3461740">the guy</span> <span m="3461800">to</span> <span m="3461980">the</span> <span m="3462050">left,</span> <span m="3462360">and the</span> <span m="3462450">guy</span> <span m="3462570">below.</span> <span m="3463460">Two</span> <span m="3463690">of</span> <span m="3463730">those</span> <span m="3463950">are</span> <span m="3464040">row</span> <span m="3464210">moves,</span> <span m="3464450">two</span> <span m="3464590">are</span> <span m="3464670">columns.</span> <span m="3465980">So</span> <span m="3465990">that's</span> <span m="3466220">right--</span> <span m="3466390">to</span> <span m="3466490">really</span> <span m="3466760">pin</span> <span m="3466940">down</span> <span m="3467110">the</span> <span m="3467190">details,</span> <span m="3467590">we</span> <span m="3467650">should</span> <span m="3467810">have</span> <span m="3467910">checked</span> <span m="3468830">those</span> <span m="3469140">are</span> <span m="3469170">the</span> <span m="3469300">only</span> <span m="3469470">two</span> <span m="3469610">possible</span> <span m="3470060">moves.</span> </p>
<p><span m="3471160">And</span> <span m="3471340">if</span> <span m="3471420">there</span> <span m="3471520">was</span> <span m="3471650">a</span> <span m="3471720">third</span> <span m="3472010">kind</span> <span m="3472200">of</span> <span m="3472250">move</span> <span m="3472450">that</span> <span m="3472550">I</span> <span m="3472620">hadn't</span> <span m="3472850">considered,</span> <span m="3473320">this</span> <span m="3473470">proof</span> <span m="3473690">would</span> <span m="3473780">be</span> <span m="3473890">bogus.</span> <span m="3475860">That's</span> <span m="3476070">a</span> <span m="3476110">good</span> <span m="3476240">question.</span> <span m="3478290">Any</span> <span m="3478530">other</span> <span m="3478700">questions?</span> <span m="3482230">OK.</span> </p>
<p><span m="3484880">Now,</span> <span m="3486060">I</span> <span m="3486130">think</span> <span m="3486380">probably, you</span> <span m="3486850">see</span> <span m="3487040">why</span> <span m="3487230">the</span> <span m="3487360">invariants</span> <span m="3487860">are</span> <span m="3487910">important.</span> <span m="3488350">But</span> <span m="3488940">say</span> <span m="3489980">that</span> <span m="3490190">someday,</span> <span m="3490530">you're</span> <span m="3490740">building</span> <span m="3491200">software</span> <span m="3492290">to</span> <span m="3492420">run</span> <span m="3492640">a</span> <span m="3492690">nuclear</span> <span m="3493070">reactor.</span> <span m="3495000">Now,</span> <span m="3495080">there's</span> <span m="3495240">a</span> <span m="3495310">certain</span> <span m="3495620">state</span> <span m="3495970">you'd</span> <span m="3496150">really</span> <span m="3496470">like</span> <span m="3496690">to</span> <span m="3496800">avoid--</span> <span m="3497970">meltdown.</span> <span m="3499530">And</span> <span m="3499810">you'd</span> <span m="3500000">like</span> <span m="3500220">to</span> <span m="3500310">be</span> <span m="3500410">able</span> <span m="3500550">to</span> <span m="3500620">prove</span> <span m="3501310">that,</span> <span m="3501670">in</span> <span m="3501820">any</span> <span m="3502100">sequence</span> <span m="3502660">of</span> <span m="3502710">moves</span> <span m="3503180">that</span> <span m="3503270">happen</span> <span m="3504420">in</span> <span m="3504580">your</span> <span m="3504770">reactor</span> <span m="3505220">program,</span> <span m="3506020">you</span> <span m="3506150">never</span> <span m="3506470">reach</span> <span m="3506800">the</span> <span m="3506880">meltdown</span> <span m="3507320">state.</span> </p>
<p><span m="3508800">Well</span> <span m="3508960">you'd</span> <span m="3509140">use</span> <span m="3509310">invariants</span> <span m="3509950">to</span> <span m="3510060">do</span> <span m="3510160">that.</span> <span m="3511160">Or</span> <span m="3511560">if</span> <span m="3511630">you're</span> <span m="3511760">building</span> <span m="3513000">software</span> <span m="3513560">for</span> <span m="3513720">an airplane,</span> <span m="3514950">you'd</span> <span m="3515110">like</span> <span m="3515250">to</span> <span m="3515350">never</span> <span m="3515650">reach</span> <span m="3515980">the</span> <span m="3516060">crash</span> <span m="3516450">state.</span> <span m="3517400">Or</span> <span m="3517730">if</span> <span m="3517790">you're</span> <span m="3517910">building</span> <span m="3518260">a</span> <span m="3518320">radiation</span> <span m="3518910">device,</span> <span m="3519860">you</span> <span m="3519980">never</span> <span m="3520210">want</span> <span m="3520400">to</span> <span m="3520450">get</span> <span m="3520610">in</span> <span m="3520680">the</span> <span m="3520760">state</span> <span m="3521300">that</span> <span m="3521790">fries</span> <span m="3522350">the</span> <span m="3522430">patient.</span> <span m="3524330">So</span> <span m="3525350">this</span> <span m="3525620">is</span> <span m="3525730">an</span> <span m="3526120">important</span> <span m="3526590">notion.</span> </p>
<p><span m="3528610">So</span> <span m="3528790">for</span> <span m="3528920">the</span> <span m="3529010">rest</span> <span m="3529250">of</span> <span m="3529310">the</span> <span m="3529390">day,</span> <span m="3529670">we're</span> <span m="3529800">going</span> <span m="3529890">to</span> <span m="3529950">talk</span> <span m="3530230">about</span> <span m="3530500">a</span> <span m="3530550">different</span> <span m="3531050">kind</span> <span m="3531280">of</span> <span m="3531360">induction</span> <span m="3532470">called</span> <span m="3532890">strong</span> <span m="3533370">induction.</span> <span m="3535390">It's</span> <span m="3535610">very</span> <span m="3535950">similar</span> <span m="3536350">to</span> <span m="3536430">ordinary</span> <span m="3536950">induction,</span> <span m="3538020">but</span> <span m="3538250">it's</span> <span m="3538370">a</span> <span m="3538420">little</span> <span m="3538760">easier</span> <span m="3539210">to</span> <span m="3539260">use</span> <span m="3540020">when</span> <span m="3540170">solving</span> <span m="3540770">certain</span> <span m="3541190">problems.</span> <span m="3542970">Now</span> <span m="3543750">like</span> <span m="3544060">regular</span> <span m="3544420">induction,</span> <span m="3545010">strong</span> <span m="3545380">induction</span> <span m="3545820">can</span> <span m="3545970">be</span> <span m="3546070">expressed</span> <span m="3547220">with</span> <span m="3547370">an</span> <span m="3547450">axiom.</span> <span m="3552490">So</span> <span m="3552650">let</span> <span m="3552750">me</span> <span m="3552850">show</span> <span m="3553080">you</span> <span m="3553210">the</span> <span m="3553270">strong</span> <span m="3553600">induction</span> <span m="3554097">axiom.</span> </p>
<p></p>
<p><span m="3584420">You</span> <span m="3584580">have</span> <span m="3584790">a</span> <span m="3584870">predicate,</span> <span m="3585380">P n,</span> <span m="3585780">like</span> <span m="3585950">before.</span> <span m="3594570">If</span> <span m="3595130">P0</span> <span m="3595870">is</span> <span m="3595990">true--</span> <span m="3596450">your</span> <span m="3596570">base</span> <span m="3596860">case--</span> <span m="3601870">and</span> <span m="3602260">for</span> <span m="3602490">all</span> <span m="3602720">n,</span> <span m="3604840">it's</span> <span m="3605000">not</span> <span m="3605250">P n</span> <span m="3605610">implies</span> <span m="3606130">P n</span> <span m="3606420">plus</span> <span m="3606640">1,</span> <span m="3606880">it's</span> <span m="3607040">P0</span> <span m="3608290">and</span> <span m="3608940">P1</span> <span m="3610330">and</span> <span m="3612010">dot,</span> <span m="3612150">dot, dot,</span> <span m="3612410">and</span> <span m="3612820">P n</span> <span m="3613560">are</span> <span m="3613740">all</span> <span m="3614020">true,</span> <span m="3615580">then</span> <span m="3616110">P n</span> <span m="3616450">plus</span> <span m="3616700">1.</span> <span m="3619430">If</span> <span m="3619920">this</span> <span m="3620310">is</span> <span m="3620420">true</span> <span m="3620810">for</span> <span m="3621020">all</span> <span m="3621280">n,</span> <span m="3627400">then</span> <span m="3627860">P n</span> <span m="3628210">is</span> <span m="3628300">true</span> <span m="3628450">for</span> <span m="3628610">all</span> <span m="3628760">n.</span> </p>
<p><span m="3638410">Now,</span> <span m="3638430">the</span> <span m="3638580">only</span> <span m="3638890">difference</span> <span m="3639320">between</span> <span m="3639590">strong</span> <span m="3640030">induction</span> <span m="3641470">and</span> <span m="3641720">ordinary</span> <span m="3642470">induction</span> <span m="3642725">that</span> <span m="3642980">we</span> <span m="3643090">did</span> <span m="3643250">last</span> <span m="3643550">time</span> <span m="3645250">is</span> <span m="3645720">this</span> <span m="3645920">part.</span> <span m="3649460">In</span> <span m="3649620">ordinary</span> <span m="3650120">induction,</span> <span m="3651820">you're</span> <span m="3651850">showing</span> <span m="3652220">that</span> <span m="3652330">P n</span> <span m="3652800">implies</span> <span m="3653230">P n</span> <span m="3653520">plus</span> <span m="3653820">1.</span> <span m="3654830">In</span> <span m="3654970">strong</span> <span m="3655440">induction,</span> <span m="3656760">you're</span> <span m="3656800">showing</span> <span m="3657210">that all</span> <span m="3657680">these</span> <span m="3657960">facts</span> <span m="3658360">put</span> <span m="3658550">together</span> <span m="3659850">imply</span> <span m="3660250">P n</span> <span m="3660620">plus</span> <span m="3660880">1.</span> <span m="3662880">Now,</span> <span m="3663140">to</span> <span m="3663230">show</span> <span m="3663640">implication,</span> <span m="3664390">remember,</span> <span m="3664810">you</span> <span m="3664930">get</span> <span m="3665110">to</span> <span m="3665210">assume</span> <span m="3665670">that</span> <span m="3665800">all</span> <span m="3666280">these</span> <span m="3666540">things</span> <span m="3666800">are</span> <span m="3666870">true.</span> </p>
<p><span m="3668520">So</span> <span m="3668640">you</span> <span m="3668680">know</span> <span m="3668810">how</span> <span m="3668970">in</span> <span m="3669060">ordinary</span> <span m="3669520">induction, we</span> <span m="3669840">say,</span> <span m="3670150">"assume</span> <span m="3670590">P n</span> <span m="3670900">is</span> <span m="3671000">true</span> <span m="3671360">and</span> <span m="3671720">you're</span> <span m="3671820">going</span> <span m="3671910">to</span> <span m="3671960">prove</span> <span m="3672240">P n</span> <span m="3672490">plus</span> <span m="3672720">1?"</span> <span m="3673890">In</span> <span m="3674010">strong</span> <span m="3674450">induction,</span> <span m="3674900">you</span> <span m="3674990">get</span> <span m="3675140">to</span> <span m="3675230">say,</span> <span m="3676170">"assume</span> <span m="3676690">P n is</span> <span m="3677180">true,</span> <span m="3677440">P n</span> <span m="3677750">minus</span> <span m="3678080">1</span> <span m="3678280">is</span> <span m="3678380">true,</span> <span m="3678740">dot,</span> <span m="3679070">dot, dot,</span> <span m="3679400">P1 is</span> <span m="3679860">true,</span> <span m="3680070">and P0</span> <span m="3680240">is</span> <span m="3680530">true.</span> <span m="3681840">You</span> <span m="3681960">get</span> <span m="3682150">to</span> <span m="3682240">assume</span> <span m="3682680">a</span> <span m="3682750">lot</span> <span m="3683170">more.</span> <span m="3684650">So</span> <span m="3684840">it's</span> <span m="3684930">a</span> <span m="3685000">stronger</span> <span m="3685840">proof</span> <span m="3686120">technique,</span> <span m="3686810">because</span> <span m="3687040">it</span> <span m="3687100">allows</span> <span m="3687510">you</span> <span m="3687750">to</span> <span m="3687820">do</span> <span m="3687960">more--</span> <span m="3688380">or</span> <span m="3688640">so</span> <span m="3688850">you</span> <span m="3688980">would</span> <span m="3689100">think.</span> </p>
<p><span m="3690520">Now</span> <span m="3690690">in</span> <span m="3690770">fact,</span> <span m="3692310">any</span> <span m="3692600">proof</span> <span m="3692900">you</span> <span m="3693010">can</span> <span m="3693180">do</span> <span m="3693370">with</span> <span m="3693480">strong</span> <span m="3693790">induction,</span> <span m="3694190">you</span> <span m="3694280">can</span> <span m="3694420">do</span> <span m="3694520">with</span> <span m="3694640">ordinary</span> <span m="3695020">induction.</span> <span m="3695620">It</span> <span m="3695710">just</span> <span m="3695900">might</span> <span m="3696020">be</span> <span m="3696120">harder.</span> <span m="3696990">So</span> <span m="3697020">you</span> <span m="3697110">can't</span> <span m="3697440">prove</span> <span m="3697690">any more</span> <span m="3698090">with</span> <span m="3698250">it,</span> <span m="3698600">but</span> <span m="3698930">it</span> <span m="3699000">makes</span> <span m="3699200">your</span> <span m="3699310">proofs</span> <span m="3699600">much</span> <span m="3699870">easier.</span> </p>
<p><span m="3700300">And</span> <span m="3700400">we're</span> <span m="3700500">going</span> <span m="3700600">to</span> <span m="3700700">do</span> <span m="3701100">an</span> <span m="3701170">example</span> <span m="3701590">in</span> <span m="3701640">a</span> <span m="3701690">minute</span> <span m="3701930">where</span> <span m="3702000">the</span> <span m="3702100">proof</span> <span m="3702300">is</span> <span m="3702410">much</span> <span m="3702700">easier</span> <span m="3703760">by</span> <span m="3703900">getting</span> <span m="3704320">to</span> <span m="3704460">assume</span> <span m="3704980">all</span> <span m="3705480">of</span> <span m="3705600">these</span> <span m="3705860">things</span> <span m="3706110">are</span> <span m="3706190">true</span> <span m="3707040">in</span> <span m="3707150">order</span> <span m="3707330">to</span> <span m="3707400">prove</span> <span m="3707640">P</span> <span m="3707830">n</span> <span m="3707910">plus</span> <span m="3708140">1.</span> <span m="3710880">Are</span> <span m="3710930">there</span> <span m="3711040">any</span> <span m="3711170">questions</span> <span m="3711620">about</span> <span m="3713220">the</span> <span m="3713330">difference</span> <span m="3713800">here</span> <span m="3713990">with</span> <span m="3714070">strong</span> <span m="3714360">induction--</span> <span m="3715480">that</span> <span m="3715670">now</span> <span m="3715900">you</span> <span m="3716020">get</span> <span m="3716150">to</span> <span m="3716220">assume</span> <span m="3716590">all</span> <span m="3716910">these</span> <span m="3717150">are</span> <span m="3717230">true</span> <span m="3718530">as</span> <span m="3718650">part</span> <span m="3718830">of the</span> <span m="3718870">inductive</span> <span m="3719280">step?</span> <span m="3722000">All</span> <span m="3722090">right.</span> </p>
<p><span m="3724930">Our</span> <span m="3725010">first</span> <span m="3725300">example</span> <span m="3725600">of</span> <span m="3725900">a</span> <span m="3726000">strong</span> <span m="3726190">induction</span> <span m="3727410">is</span> <span m="3727620">going</span> <span m="3727730">to</span> <span m="3727780">be</span> <span m="3728280">a</span> <span m="3728360">simple</span> <span m="3728700">game.</span> <span m="3730210">We've</span> <span m="3730370">got</span> <span m="3730490">a</span> <span m="3730540">stack</span> <span m="3730960">of</span> <span m="3731090">eight</span> <span m="3731310">blocks</span> <span m="3731700">here.</span> <span m="3733380">Now,</span> <span m="3733540">in</span> <span m="3733610">this</span> <span m="3733820">game,</span> <span m="3734720">what</span> <span m="3735020">you're</span> <span m="3735130">going</span> <span m="3735250">to</span> <span m="3735320">do</span> <span m="3735680">is</span> <span m="3735880">divide</span> <span m="3737170">the</span> <span m="3737290">stack</span> <span m="3738580">into</span> <span m="3739770">two</span> <span m="3739960">sub-stacks.</span> <span m="3741490">So</span> <span m="3741670">I</span> <span m="3741770">just</span> <span m="3741990">took</span> <span m="3742450">eight,</span> <span m="3742790">and</span> <span m="3742910">split</span> <span m="3743170">it</span> <span m="3743260">into</span> <span m="3743450">three</span> <span m="3743770">and</span> <span m="3743880">five.</span> </p>
<p><span m="3745470">For</span> <span m="3745610">that</span> <span m="3745830">move,</span> <span m="3746200">you</span> <span m="3746320">get</span> <span m="3746560">15</span> <span m="3747030">points--</span> <span m="3747982">3</span> <span m="3748460">times</span> <span m="3748790">5.</span> <span m="3750390">Let's write</span> <span m="3750500">that</span> <span m="3750995">down.</span> </p>
<p></p>
<p><span m="3765370">OK,</span> <span m="3765660">so</span> <span m="3765810">this</span> <span m="3766150">is</span> <span m="3766300">called</span> <span m="3766480">the</span> <span m="3766590">Unstacking</span> <span m="3767220">Game.</span> <span m="3774810">And</span> <span m="3775080">in the</span> <span m="3775150">first</span> <span m="3775550">move,</span> <span m="3776040">we</span> <span m="3776170">went</span> <span m="3777590">8</span> <span m="3778540">split</span> <span m="3779360">to</span> <span m="3780550">5</span> <span m="3781050">and</span> <span m="3781180">3.</span> <span m="3782140">And</span> <span m="3782280">that's</span> <span m="3782550">worth</span> <span m="3782910">15</span> <span m="3783450">points.</span> </p>
<p><span m="3786850">Now</span> <span m="3787110">in</span> <span m="3787170">the</span> <span m="3787250">next</span> <span m="3787560">move,</span> <span m="3790270">I'm</span> <span m="3790420">going</span> <span m="3790540">to</span> <span m="3790630">split</span> <span m="3791180">the</span> <span m="3791310">stack</span> <span m="3791640">of</span> <span m="3791740">five</span> <span m="3792100">into</span> <span m="3792260">four</span> <span m="3792660">and</span> <span m="3792750">one.</span> <span m="3794160">And</span> <span m="3794350">for</span> <span m="3794430">that,</span> <span m="3794680">you</span> <span m="3794770">get</span> <span m="3794970">4 points--</span> <span m="3795842">4</span> <span m="3796280">times</span> <span m="3796550">1.</span> <span m="3801230">And</span> <span m="3801400">I</span> <span m="3801470">keep</span> <span m="3801770">on</span> <span m="3801870">going</span> <span m="3803510">until</span> <span m="3803840">I</span> <span m="3803940">have</span> <span m="3804100">a</span> <span m="3804290">stacks</span> <span m="3804720">of</span> <span m="3804790">height</span> <span m="3804990">one.</span> </p>
<p><span m="3806610">And</span> <span m="3806770">the</span> <span m="3806850">last</span> <span m="3807230">move,</span> <span m="3807450">I'll be</span> <span m="3807950">splitting a</span> <span m="3808050">two</span> <span m="3808260">into</span> <span m="3808720">two</span> <span m="3808890">ones.</span> <span m="3809270">That'll</span> <span m="3809350">give</span> <span m="3809450">me</span> <span m="3809530">one</span> <span m="3809810">point.</span> <span m="3810810">Then</span> <span m="3810950">I</span> <span m="3811070">add</span> <span m="3811320">up</span> <span m="3811420">all</span> <span m="3811740">the</span> <span m="3811840">points,</span> <span m="3813260">and</span> <span m="3813370">that's</span> <span m="3813540">your</span> <span m="3813660">score.</span> <span m="3815080">And</span> <span m="3815240">the</span> <span m="3815320">goal</span> <span m="3815730">is</span> <span m="3815840">to</span> <span m="3815930">get</span> <span m="3816280">the</span> <span m="3816360">most</span> <span m="3817240">number</span> <span m="3817540">of</span> <span m="3817620">points</span> <span m="3818630">when</span> <span m="3818730">you</span> <span m="3818800">add</span> <span m="3818990">up</span> <span m="3819100">the</span> <span m="3819180">entire</span> <span m="3819530">score.</span> </p>
<p><span m="3821200">Now,</span> <span m="3821440">just</span> <span m="3821780">so</span> <span m="3821920">this</span> <span m="3822130">game</span> <span m="3822350">is</span> <span m="3822460">clear,</span> <span m="3823460">we're</span> <span m="3823670">going</span> <span m="3823770">to</span> <span m="3823840">play</span> <span m="3824110">it.</span> <span m="3825730">And</span> <span m="3825950">I'm</span> <span m="3826030">going</span> <span m="3826140">to</span> <span m="3826220">raise</span> <span m="3826550">the</span> <span m="3826650">stakes</span> <span m="3827080">here.</span> <span m="3828386">What</span> <span m="3828760">we're</span> <span m="3828960">going</span> <span m="3829060">to</span> <span m="3829130">do</span> <span m="3829340">is</span> <span m="3829440">we're</span> <span m="3829510">going</span> <span m="3829590">to</span> <span m="3829650">have</span> <span m="3829930">a</span> <span m="3830030">competition</span> <span m="3830960">between</span> <span m="3831450">three</span> <span m="3831660">members</span> <span m="3831980">of</span> <span m="3832020">the</span> <span m="3832100">class</span> <span m="3833500">and</span> <span m="3833750">the</span> <span m="3833830">TAs.</span> </p>
<p><span m="3835610">Now,</span> <span m="3836510">if</span> <span m="3837100">the</span> <span m="3837190">class</span> <span m="3837590">team</span> <span m="3837850">wins--</span> <span m="3838340">they</span> <span m="3838450">get</span> <span m="3838630">more</span> <span m="3838870">points--</span> <span m="3840020">we're</span> <span m="3840180">going</span> <span m="3840260">to</span> <span m="3840300">give</span> <span m="3840480">everybody</span> <span m="3840790">in the</span> <span m="3840900">class</span> <span m="3842130">candy.</span> <span m="3842720">And</span> <span m="3843320">I've</span> <span m="3843470">got</span> <span m="3843690">lots</span> <span m="3843920">of</span> <span m="3844000">candy</span> <span m="3844370">here.</span> <span m="3846340">If</span> <span m="3846510">the</span> <span m="3846630">TAs</span> <span m="3847100">beat</span> <span m="3847370">you,</span> <span m="3848460">they</span> <span m="3848620">get</span> <span m="3848750">the</span> <span m="3848840">candy.</span> </p>
<p><span m="3849668">AUDIENCE: What if it's</span> <span m="3850126">a tie?</span> </p>
<p><span m="3851610">PROFESSOR: We'll</span> <span m="3851720">figure</span> <span m="3851970">that</span> <span m="3852190">out.</span> <span m="3853790">We'll</span> <span m="3853940">come</span> <span m="3854120">to</span> <span m="3854170">that</span> <span m="3854380">later.</span> <span m="3854720">But</span> <span m="3854750">you</span> <span m="3854840">want</span> <span m="3855030">to</span> <span m="3855110">win</span> <span m="3855300">this</span> <span m="3855500">thing.</span> </p>
<p><span m="3856140">So</span> <span m="3856450">I</span> <span m="3856530">need</span> <span m="3856800">some</span> <span m="3856940">volunteers</span> <span m="3857730">from</span> <span m="3857830">the</span> <span m="3857910">class</span> <span m="3858320">who</span> <span m="3858400">think</span> <span m="3858580">they</span> <span m="3858680">can</span> <span m="3858840">do</span> <span m="3858980">well.</span> <span m="3859340">Yeah,</span> <span m="3859880">come</span> <span m="3860050">on</span> <span m="3860190">down.</span> <span m="3861030">You</span> <span m="3861190">two</span> <span m="3861310">guys</span> <span m="3861600">come</span> <span m="3861750">on</span> <span m="3861900">down.</span> <span m="3863740">All right,</span> <span m="3863980">TAs</span> <span m="3864460">come</span> <span m="3864600">on</span> <span m="3864760">up.</span> </p>
<p><span m="3870030">Now,</span> <span m="3870330">I</span> <span m="3870500">want</span> <span m="3870740">you</span> <span m="3870810">to</span> <span m="3870900">make</span> <span m="3871130">some</span> <span m="3871270">noise</span> <span m="3871760">if</span> <span m="3871880">you</span> <span m="3872090">think</span> <span m="3872400">your</span> <span m="3872570">three</span> <span m="3872840">class</span> <span m="3873210">reps</span> <span m="3873480">here</span> <span m="3874260">can</span> <span m="3874430">beat</span> <span m="3874620">the</span> <span m="3874720">TAs.</span> </p>
<p><span m="3875796">[CHEERS AND APPLAUSE]</span> </p>
<p><span m="3878102">All right,</span> <span m="3879470">how</span> <span m="3879650">many</span> <span m="3879880">people--</span> <span m="3880230">make</span> <span m="3880430">some</span> <span m="3880560">noise</span> <span m="3880830">you</span> <span m="3880970">think</span> <span m="3881140">the</span> <span m="3881220">TAs</span> <span m="3881580">are</span> <span m="3881630">going</span> <span m="3881720">to</span> <span m="3881780">win.</span> </p>
<p><span m="3882540">AUDIENCE: Ooh.</span> </p>
<p><span m="3883480">PROFESSOR: Ooh, that's</span> <span m="3883950">nasty.</span> <span m="3884420">It</span> <span m="3884890">must've</span> <span m="3885170">been</span> <span m="3885270">a</span> <span m="3885340">rough</span> <span m="3885570">recitation</span> <span m="3886240">last</span> <span m="3886510">week</span> <span m="3887180">on</span> <span m="3887280">Wednesday.</span> <span m="3888930">All</span> <span m="3889160">right,</span> <span m="3889520">so</span> <span m="3890870">let's</span> <span m="3891060">see.</span> <span m="3891160">I</span> <span m="3891230">think</span> <span m="3891390">we're</span> <span m="3891480">going</span> <span m="3891560">to</span> <span m="3891600">let</span> <span m="3891730">the</span> <span m="3891820">class</span> <span m="3892300">go</span> <span m="3892460">first.</span> </p>
<p><span m="3892800">This</span> <span m="3892980">is</span> <span m="3893060">your</span> <span m="3893220">stack.</span> <span m="3894160">Now</span> <span m="3894490">think</span> <span m="3894740">about</span> <span m="3894970">what</span> <span m="3895150">move</span> <span m="3895380">you</span> <span m="3895490">want</span> <span m="3895650">to</span> <span m="3895690">make</span> <span m="3895890">first,</span> <span m="3896270">because</span> <span m="3896780">that</span> <span m="3896970">matters.</span> <span m="3897750">And</span> <span m="3898940">you</span> <span m="3899090">guys</span> <span m="3899340">can</span> <span m="3899450">give</span> <span m="3899600">them</span> <span m="3899700">some</span> <span m="3899840">advice</span> <span m="3900130">if</span> <span m="3900220">you</span> <span m="3900330">want,</span> <span m="3900630">as</span> <span m="3900780">to</span> <span m="3900860">the</span> <span m="3900940">move</span> <span m="3901180">they</span> <span m="3901270">should</span> <span m="3901470">make</span> <span m="3901650">first.</span> <span m="3903010">And</span> <span m="3903120">I'm</span> <span m="3903200">going</span> <span m="3903310">to</span> <span m="3903370">keep</span> <span m="3903560">score</span> <span m="3903880">over</span> <span m="3904090">here.</span> </p>
<p><span m="3908820">Oh,</span> <span m="3909210">OK,</span> <span m="3909570">they</span> <span m="3909760">went</span> <span m="3909940">for</span> <span m="3910000">the</span> <span m="3910090">maximum</span> <span m="3910810">move.</span> <span m="3911070">That's</span> <span m="3911370">good.</span> <span m="3911580">They</span> <span m="3911720">realized</span> <span m="3912690">4</span> <span m="3913020">times</span> <span m="3913360">4</span> <span m="3913700">is</span> <span m="3913800">16--</span> <span m="3915470">very</span> <span m="3915800">good.</span> <span m="3916160">So</span> <span m="3916300">let's</span> <span m="3916470">write</span> <span m="3916690">that</span> <span m="3916900">up</span> <span m="3917060">here.</span> </p>
<p><span m="3919705">You're</span> <span m="3920160">Off</span> <span m="3920360">to</span> <span m="3920490">a</span> <span m="3920540">good</span> <span m="3920720">start.</span> <span m="3923380">In</span> <span m="3923590">fact,</span> <span m="3923810">that's</span> <span m="3923990">the</span> <span m="3924080">maximum</span> <span m="3924760">number</span> <span m="3924970">of</span> <span m="3925050">points</span> <span m="3925370">for</span> <span m="3925450">the</span> <span m="3925530">first</span> <span m="3925840">move,</span> <span m="3926080">I</span> <span m="3926180">think--</span> <span m="3928300">good</span> <span m="3928450">job.</span> <span m="3932320">All</span> <span m="3932500">right,</span> <span m="3932720">so</span> <span m="3932810">here's</span> <span m="3933050">the</span> <span m="3933130">class</span> <span m="3933770">here.</span> <span m="3936490">And</span> <span m="3936980">here are</span> <span m="3937240">the</span> <span m="3937390">TAs.</span> </p>
<p><span m="3940620">And</span> <span m="3940770">the</span> <span m="3940850">class</span> <span m="3941370">went</span> <span m="3942990">8</span> <span m="3943370">into</span> <span m="3944650">4</span> <span m="3945210">by</span> <span m="3945360">4,</span> <span m="3946920">and</span> <span m="3947210">they</span> <span m="3947360">got</span> <span m="3947800">16.</span> <span m="3951030">Now, the</span> <span m="3951190">TAs</span> <span m="3951670">are</span> <span m="3951730">working</span> <span m="3952030">hard</span> <span m="3952330">here,</span> <span m="3952540">as</span> <span m="3952590">you</span> <span m="3952710">can</span> <span m="3952840">see.</span> <span m="3955340">And</span> <span m="3955420">they've</span> <span m="3955520">got</span> <span m="3955700">a</span> <span m="3955740">move.</span> <span m="3956890">That's</span> <span m="3957150">a</span> <span m="3957240">pretty</span> <span m="3957490">pathetic</span> <span m="3957920">first</span> <span m="3958200">move</span> <span m="3959230">for</span> <span m="3959300">the</span> <span m="3959410">TAs.</span> <span m="3959910">Looking</span> <span m="3960220">good</span> <span m="3960470">for</span> <span m="3960560">the</span> <span m="3960650">candy</span> <span m="3961040">for</span> <span m="3961130">you</span> <span m="3961250">guys.</span> <span m="3963230">They</span> <span m="3963400">got</span> <span m="3963610">7</span> <span m="3964020">points</span> <span m="3965490">for the</span> <span m="3965620">TAs.</span> </p>
<p><span m="3966980">Class,</span> <span m="3967840">what</span> <span m="3968010">would</span> <span m="3968110">you</span> <span m="3968240">like</span> <span m="3968430">to</span> <span m="3968530">do</span> <span m="3968710">for</span> <span m="3968820">your</span> <span m="3968920">next</span> <span m="3969220">move?</span> </p>
<p><span m="3970119">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="3976220">PROFESSOR: You</span> <span m="3976510">can't</span> <span m="3976850">start</span> <span m="3977180">over</span> <span m="3977420">now--</span> <span m="3977930">too</span> <span m="3978130">late</span> <span m="3978410">for</span> <span m="3978510">that.</span> </p>
<p><span m="3979274">[SIDE CONVERSATION]</span> </p>
<p><span m="3982000">A</span> <span m="3982340">lot</span> <span m="3982520">of</span> <span m="3982560">people</span> <span m="3982810">counting</span> <span m="3983120">on</span> <span m="3983280">you</span> <span m="3983400">guys.</span> </p>
<p><span m="3984140">[SIDE CONVERSATION]</span> </p>
<p><span m="3987830">Oh,</span> <span m="3988250">no,</span> <span m="3988270">no,</span> <span m="3988720">one</span> <span m="3988980">at</span> <span m="3989170">a</span> <span m="3989240">time.</span> <span m="3989480">No,</span> <span m="3989690">just</span> <span m="3989890">one.</span> <span m="3990990">One</span> <span m="3991360">stack</span> <span m="3991600">gets</span> <span m="3991770">split--</span> <span m="3992450">either</span> <span m="3992690">one.</span> </p>
<p><span m="3993866">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="3995540">PROFESSOR: There</span> <span m="3995800">you</span> <span m="3995900">go.</span> <span m="3997200">Oh,</span> <span m="3997540">well,</span> <span m="3998150">what do</span> <span m="3998420">you</span> <span m="3998800">do</span> <span m="3998940">now?</span> </p>
<p><span m="3999668">[SIDE CONVERSATION]</span> </p>
<p><span m="4002656">AUDIENCE: Their strategy's</span> <span m="4003154">obviously</span> <span m="4003652">better than</span> <span m="4004150">ours.</span> </p>
<p><span m="4005146">AUDIENCE: Yeah, but we've</span> <span m="4005644">figured  it out already.</span> </p>
<p><span m="4006142">AUDIENCE: We're going to lose.</span> </p>
<p><span m="4006640">PROFESSOR: The</span> <span m="4006760">TAs</span> <span m="4007120">are</span> <span m="4007450">convinced</span> <span m="4007880">they're</span> <span m="4007990">going</span> <span m="4008090">to</span> <span m="4008150">lose.</span> <span m="4008520">Three and</span> <span m="4008850">one,</span> <span m="4009230">OK,</span> <span m="4009630">good,</span> <span m="4010700">interesting.</span> <span m="4012390">So</span> <span m="4012510">we</span> <span m="4012600">have</span> <span m="4012750">three</span> <span m="4013000">and</span> <span m="4013130">one.</span> <span m="4013320">That</span> <span m="4013480">gives</span> <span m="4013640">you</span> <span m="4013760">3</span> <span m="4014070">points.</span> </p>
<p><span m="4016110">All right,</span> <span m="4016760">TAs,</span> <span m="4017450">we</span> <span m="4017560">need</span> <span m="4017670">a</span> <span m="4017740">move.</span> <span m="4022090">All</span> <span m="4022600">right,</span> <span m="4023050">so</span> <span m="4023150">they've</span> <span m="4023330">gone</span> <span m="4024080">six</span> <span m="4024440">and</span> <span m="4024580">one--</span> <span m="4024940">6</span> <span m="4025270">points.</span> <span m="4028000">Class,</span> <span m="4031990">you</span> <span m="4032160">think</span> <span m="4032360">you're</span> <span m="4032460">dead</span> <span m="4032700">already?</span> <span m="4033330">So</span> <span m="4033420">you've</span> <span m="4033600">gotten</span> <span m="4033960">three</span> <span m="4034420">and</span> <span m="4034500">one</span> <span m="4034860">now,</span> <span m="4035220">again,</span> <span m="4036170">for</span> <span m="4036280">3.</span> </p>
<p><span m="4037130">So you're</span> <span m="4037440">up to</span> <span m="4037580">22.</span> <span m="4038160">You're</span> <span m="4038310">way</span> <span m="4038570">ahead--</span> <span m="4038870">22</span> <span m="4039280">to</span> <span m="4039370">13.</span> <span m="4040570">All</span> <span m="4040730">right,</span> <span m="4041160">TAs.</span> <span m="4043570">They</span> <span m="4043680">look</span> <span m="4043840">a</span> <span m="4043890">little</span> <span m="4044060">disorganized</span> <span m="4044680">to</span> <span m="4044750">me.</span> <span m="4044890">What</span> <span m="4045010">do</span> <span m="4045050">you</span> <span m="4045150">think?</span> </p>
<p><span m="4046130">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="4049970">PROFESSOR: Another</span> <span m="4050320">one--</span> <span m="4050420">you</span> <span m="4050490">can</span> <span m="4050620">tell</span> <span m="4050870">where</span> <span m="4051150">they're headed</span> <span m="4051480">here--</span> <span m="4052390">five</span> <span m="4052720">and</span> <span m="4052790">one.</span> <span m="4052970">They</span> <span m="4053060">have</span> <span m="4053200">5</span> <span m="4053520">points.</span> <span m="4053850">They're up to</span> <span m="4054140">18.</span> <span m="4054500">They're 4</span> <span m="4054860">behind</span> <span m="4055220">you.</span> </p>
<p><span m="4057300">All</span> <span m="4057410">right,</span> <span m="4057540">class.</span> <span m="4059146">All right,</span> <span m="4061480">you</span> <span m="4061610">got</span> <span m="4061840">2</span> <span m="4062020">points.</span> <span m="4063640">TAs--</span> <span m="4071180">you</span> <span m="4071530">really</span> <span m="4071760">think</span> <span m="4071950">they</span> <span m="4072070">know</span> <span m="4072280">how</span> <span m="4072390">to</span> <span m="4072500">do</span> <span m="4072650">this</span> <span m="4072950">best?</span> <span m="4073290">I</span> <span m="4073350">don't</span> <span m="4073520">know.</span> </p>
<p><span m="4074295">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="4082380">PROFESSOR: Oh,</span> <span m="4083280">big</span> <span m="4083580">change</span> <span m="4083960">in</span> <span m="4084050">strategy</span> <span m="4084620">there.</span> <span m="4088500">All</span> <span m="4088810">that</span> <span m="4089070">thinking,</span> <span m="4089520">and</span> <span m="4089620">that's</span> <span m="4089860">what</span> <span m="4090060">we</span> <span m="4090190">got.</span> <span m="4090520">OK,</span> <span m="4091310">4</span> <span m="4092020">points.</span> <span m="4093406">Class.</span> <span m="4100029">All right,</span> <span m="4100560">they</span> <span m="4100670">pick</span> <span m="4100930">up</span> <span m="4101120">2</span> <span m="4101410">more</span> <span m="4102149">here.</span> <span m="4102939">The</span> <span m="4103060">three</span> <span m="4103430">just</span> <span m="4103649">got</span> <span m="4103850">split</span> <span m="4104140">into</span> <span m="4105229">two</span> <span m="4105450">and</span> <span m="4105560">one</span> <span m="4106810">for</span> <span m="4107130">2</span> <span m="4107319">points.</span> </p>
<p><span m="4107920">How</span> <span m="4107990">are you</span> <span m="4108109">doing?</span> <span m="4108370">You've</span> <span m="4108439">got</span> <span m="4108590">19,</span> <span m="4109374">22,</span> <span m="4110380">24,</span> <span m="4110859">up to</span> <span m="4110970">26.</span> <span m="4112729">13,</span> <span m="4113224">18,</span> <span m="4113720">22--</span> <span m="4114520">you're</span> <span m="4114760">still</span> <span m="4114979">doing</span> <span m="4115200">good.</span> <span m="4117010">TAs</span> <span m="4117670">are</span> <span m="4117720">behind.</span> <span m="4118180">Let's</span> <span m="4118340">see</span> <span m="4118439">what</span> <span m="4118540">they've</span> <span m="4118649">got</span> <span m="4118840">up</span> <span m="4118950">their</span> <span m="4119090">sleeves</span> <span m="4119399">here.</span> </p>
<p><span m="4123210">All right,</span> <span m="4123380">they</span> <span m="4123580">definitely</span> <span m="4123939">have</span> <span m="4124120">a</span> <span m="4124240">strategy</span> <span m="4124729">going</span> <span m="4125020">here.</span> <span m="4125950">That's</span> <span m="4126270">3</span> <span m="4126550">points.</span> <span m="4129109">Ooh,</span> <span m="4129700">there's</span> <span m="4129850">not</span> <span m="4130010">much</span> <span m="4130229">option</span> <span m="4130540">left</span> <span m="4130750">for</span> <span m="4130850">you</span> <span m="4130970">guys.</span> <span m="4133029">Yeah.</span> </p>
<p><span m="4133851">[SIDE CONVERSATION]</span> </p>
<p><span m="4134813">AUDIENCE: Choose the right.</span> </p>
<p><span m="4137699">AUDIENCE: Air blocks.</span> </p>
<p><span m="4139149">PROFESSOR: No,</span> <span m="4139210">no,</span> <span m="4139710">no,</span> <span m="4141370">that'll be</span> <span m="4141520">a</span> <span m="4141700">0.</span> <span m="4145240">All right,</span> <span m="4145500">that</span> <span m="4145680">gave</span> <span m="4145880">you</span> <span m="4146020">a</span> <span m="4146109">point--</span> <span m="4147970">two</span> <span m="4148200">went</span> <span m="4148350">to</span> <span m="4148470">one</span> <span m="4148620">and</span> <span m="4148710">one--</span> <span m="4148969">1</span> <span m="4149229">point.</span> <span m="4151580">Not</span> <span m="4151810">a</span> <span m="4151859">lot</span> <span m="4152000">of</span> <span m="4152060">choice</span> <span m="4152380">for</span> <span m="4152460">you</span> <span m="4152590">guys</span> <span m="4152810">either,</span> <span m="4153080">I</span> <span m="4153149">guess.</span> <span m="4153710">There's</span> <span m="4153859">only</span> <span m="4153990">one</span> <span m="4154140">way</span> <span m="4154260">to</span> <span m="4154340">split</span> <span m="4154600">that</span> <span m="4154790">stack.</span> </p>
<p><span m="4158330">All right,</span> <span m="4158770">2</span> <span m="4158950">points,</span> <span m="4160830">and</span> <span m="4161069">your</span> <span m="4161180">final</span> <span m="4161540">move</span> <span m="4162010">for</span> <span m="4162100">the</span> <span m="4162189">class</span> <span m="4164120">and</span> <span m="4164270">all</span> <span m="4164529">the</span> <span m="4164620">candy.</span> <span m="4165979">You</span> <span m="4166149">get</span> <span m="4166319">1</span> <span m="4166529">more</span> <span m="4166800">point.</span> <span m="4169300">Let's</span> <span m="4169450">see</span> <span m="4169529">what</span> <span m="4169660">your</span> <span m="4169740">total</span> <span m="4170100">is.</span> <span m="4170340">You've</span> <span m="4170470">got</span> <span m="4170770">22,</span> <span m="4171678">24,</span> <span m="4172132">26,</span> <span m="4172586">27,</span> <span m="4173040">28.</span> <span m="4174670">And</span> <span m="4175100">the</span> <span m="4175180">TAs</span> <span m="4175830">are</span> <span m="4175930">going</span> <span m="4176040">to</span> <span m="4176120">get</span> <span m="4176689">one</span> <span m="4176830">last</span> <span m="4177160">move</span> <span m="4177410">for the</span> <span m="4177520">TAs.</span> </p>
<p><span m="4181220">All</span> <span m="4181430">right,</span> <span m="4181729">and</span> <span m="4181840">you</span> <span m="4182000">got</span> <span m="4183189">2,</span> <span m="4183479">1,</span> <span m="4183939">1,</span> <span m="4184359">2.</span> <span m="4184770">That's</span> <span m="4184950">interesting.</span> <span m="4185130">1, 2,</span> <span m="4185624">3, 4, 5,</span> <span m="4186118">6, 7--</span> <span m="4187899">7</span> <span m="4188740">times</span> <span m="4189229">8</span> <span m="4190700">over</span> <span m="4191140">2--</span> <span m="4193899">no,</span> <span m="4194100">that's</span> <span m="4194320">not</span> <span m="4194490">right.</span> <span m="4195682">Did I</span> <span m="4196050">do that</span> <span m="4196210">right?</span> <span m="4198140">Oh,</span> <span m="4198280">28--</span> <span m="4199620">we</span> <span m="4199710">got</span> <span m="4199820">a</span> <span m="4199920">tie.</span> <span m="4201800">Oh,</span> <span m="4201870">geez.</span> </p>
<p><span m="4204650">I</span> <span m="4204760">tell</span> <span m="4204910">you what--</span> <span m="4205030">I'll</span> <span m="4205240">give</span> <span m="4205370">you</span> <span m="4205450">guys</span> <span m="4205660">one</span> <span m="4205850">more</span> <span m="4206040">chance</span> <span m="4207620">to</span> <span m="4207700">start</span> <span m="4208040">over.</span> <span m="4208220">You've</span> <span m="4208340">seen</span> <span m="4208610">that.</span> <span m="4208840">You've</span> <span m="4208940">got</span> <span m="4209080">to</span> <span m="4209150">beat</span> <span m="4209840">them.</span> <span m="4210970">You tied them.</span> <span m="4211520">You've</span> <span m="4211610">got</span> <span m="4211750">to</span> <span m="4211810">beat them.</span> <span m="4212840">One</span> <span m="4213040">more</span> <span m="4213200">chance</span> <span m="4213560">here.</span> </p>
<p><span m="4214360">[SIDE CONVERSATION]</span> </p>
<p><span m="4216980">You</span> <span m="4217110">got</span> <span m="4217360">to</span> <span m="4217440">beat them, though.</span> <span m="4217680">You</span> <span m="4218010">got</span> <span m="4218170">28.</span> <span m="4218660">You</span> <span m="4218760">already</span> <span m="4218990">tied them.</span> </p>
<p><span m="4219490">They</span> <span m="4219600">came</span> <span m="4219890">from</span> <span m="4220020">behind</span> <span m="4220910">and</span> <span m="4221070">tied</span> <span m="4221420">you.</span> <span m="4222596">You've</span> <span m="4223040">got</span> <span m="4223220">to</span> <span m="4223270">beat</span> <span m="4223560">them.</span> <span m="4225380">You</span> <span m="4225490">got</span> <span m="4225760">off to</span> <span m="4225860">a</span> <span m="4225900">good</span> <span m="4226050">start,</span> <span m="4226490">but</span> <span m="4226690">then</span> <span m="4226870">they</span> <span m="4227000">caught</span> <span m="4227270">up.</span> </p>
<p><span m="4227730">[SIDE CONVERSATION]</span> </p>
<p><span m="4237720">Now,</span> <span m="4237850">your</span> <span m="4238500">class</span> <span m="4238820">reps</span> <span m="4239140">had</span> <span m="4239280">an</span> <span m="4239410">interesting</span> <span m="4239860">thought</span> <span m="4240200">here.</span> <span m="4241210">They</span> <span m="4241350">ask,</span> <span m="4241650">is</span> <span m="4241810">it</span> <span m="4241960">possible</span> <span m="4242540">that</span> <span m="4242690">any</span> <span m="4242990">strategy</span> <span m="4243640">gives</span> <span m="4243920">28?</span> <span m="4247160">Hmm.</span> <span m="4249590">I</span> <span m="4249740">wouldn't</span> <span m="4250050">do</span> <span m="4250200">that</span> <span m="4250450">to</span> <span m="4250520">you,</span> <span m="4250690">would</span> <span m="4250900">I?</span> </p>
<p><span m="4251110">[LAUGHTER]</span> </p>
<p><span m="4253082">Yeah,</span> <span m="4254070">probably.</span> <span m="4256670">Any</span> <span m="4256910">ideas?</span> <span m="4257545">Can you</span> <span m="4257990">beat</span> <span m="4258435">28?</span> <span m="4261110">They're</span> <span m="4261320">already</span> <span m="4261610">eating</span> <span m="4261730">the</span> <span m="4262040">candy.</span> </p>
<p><span m="4266770">In</span> <span m="4267130">fact,</span> <span m="4268550">you</span> <span m="4268650">can't</span> <span m="4268990">beat</span> <span m="4269150">28.</span> <span m="4270370">And</span> <span m="4270530">you</span> <span m="4270610">can't</span> <span m="4270850">do</span> <span m="4270970">worse</span> <span m="4271290">than</span> <span m="4271440">28,</span> <span m="4271725">either.</span> <span m="4272310">Because</span> <span m="4272690">every</span> <span m="4273040">strategy</span> <span m="4273510">gives</span> <span m="4273690">28.</span> <span m="4274780">So</span> <span m="4274980">I</span> <span m="4275060">guess</span> <span m="4275320">we're</span> <span m="4275420">going</span> <span m="4275510">to</span> <span m="4275560">have</span> <span m="4275760">to</span> <span m="4275880">throw</span> <span m="4276110">it to</span> <span m="4276240">the</span> <span m="4276320">class</span> <span m="4276690">to</span> <span m="4276780">vote</span> <span m="4277630">who</span> <span m="4277900">should</span> <span m="4278100">get</span> <span m="4278280">the</span> <span m="4278370">candy.</span> <span m="4280270">So</span> <span m="4280530">if</span> <span m="4280720">you</span> <span m="4280860">think</span> <span m="4281410">your</span> <span m="4281680">reps</span> <span m="4282540">really</span> <span m="4282890">should</span> <span m="4283090">win</span> <span m="4283300">this</span> <span m="4283510">and you</span> <span m="4283660">should</span> <span m="4283850">get the</span> <span m="4283930">candy,</span> <span m="4284250">make</span> <span m="4284440">some</span> <span m="4284550">noise.</span> </p>
<p><span m="4285090">[CHEERS AND APPLAUSE]</span> </p>
<p><span m="4288400">If you</span> <span m="4288530">think</span> <span m="4288770">your</span> <span m="4288910">TAs</span> <span m="4289850">should</span> <span m="4289950">get</span> <span m="4290140">the</span> <span m="4290230">candy</span> <span m="4290600">make</span> <span m="4290820">some</span> <span m="4290950">noise.</span> </p>
<p><span m="4291871">[CHEERS AND APPLAUSE]</span> </p>
<p><span m="4294460">All right,</span> <span m="4294810">so you get</span> <span m="4295110">the</span> <span m="4295200">candy.</span> <span m="4296380">Here,</span> <span m="4296620">help</span> <span m="4296820">pass</span> <span m="4297080">out</span> <span m="4297190">the</span> <span m="4297290">candy</span> <span m="4297610">here.</span> <span m="4297800">We've</span> <span m="4297880">got</span> <span m="4298580">plenty</span> <span m="4298810">of</span> <span m="4298870">bags</span> <span m="4299380">and</span> <span m="4299490">baskets.</span> <span m="4299970">Take</span> <span m="4300130">it</span> <span m="4300220">all</span> <span m="4300400">up</span> <span m="4300610">on</span> <span m="4300740">both</span> <span m="4300950">sides,</span> <span m="4301360">and</span> <span m="4301460">pass</span> <span m="4301620">it</span> <span m="4301710">out.</span> </p>
<p><span m="4303430">[SIDE CONVERSATION]</span> </p>
<p></p>
<p><span m="4366930">OK,</span> <span m="4367280">is</span> <span m="4367390">everybody</span> <span m="4367720">getting some</span> <span m="4368100">candy</span> <span m="4368470">up</span> <span m="4368590">there?</span> <span m="4370480">We</span> <span m="4370590">should</span> <span m="4370770">have</span> <span m="4370940">enough</span> <span m="4371140">for</span> <span m="4371260">everybody,</span> <span m="4371680">unless</span> <span m="4371890">somebody</span> <span m="4372180">got</span> <span m="4372350">really</span> <span m="4372610">hungry</span> <span m="4372930">down</span> <span m="4373160">here</span> <span m="4373350">in</span> <span m="4373450">front.</span> <span m="4379170">OK,</span> <span m="4379590">so</span> <span m="4379750">while</span> <span m="4379910">you're</span> <span m="4380040">doing</span> <span m="4380330">that,</span> <span m="4381150">let's</span> <span m="4381340">try</span> <span m="4381510">to</span> <span m="4381610">prove</span> <span m="4381920">this</span> <span m="4382120">theorem.</span> </p>
<p></p>
<p><span m="4396130">So</span> <span m="4396260">the</span> <span m="4396390">theorem</span> <span m="4396780">is</span> <span m="4396910">that</span> <span m="4396970">all</span> <span m="4397320">strategies</span> <span m="4398000">for</span> <span m="4398070">the</span> <span m="4398220">n-block</span> <span m="4398680">game</span> <span m="4401830">produce</span> <span m="4402360">the</span> <span m="4402470">same</span> <span m="4402790">score.</span> <span m="4405460">It</span> <span m="4405550">wasn't</span> <span m="4405880">a</span> <span m="4405940">coincidence</span> <span m="4406205">for</span> <span m="4406470">8.</span> <span m="4411900">And</span> <span m="4412070">the</span> <span m="4412140">score,</span> <span m="4412470">we'll</span> <span m="4412610">call</span> <span m="4412880">it</span> <span m="4413030">S</span> <span m="4413260">of</span> <span m="4413340">n</span> <span m="4413780">for</span> <span m="4413930">the</span> <span m="4414070">n-block</span> <span m="4414490">game.</span> <span m="4416070">So</span> <span m="4416160">for</span> <span m="4416280">example,</span> <span m="4419060">we're</span> <span m="4419290">trying</span> <span m="4419470">to</span> <span m="4419520">show</span> <span m="4419730">that</span> <span m="4419870">S</span> <span m="4420120">of</span> <span m="4420220">8</span> <span m="4420800">was</span> <span m="4421120">28.</span> <span m="4425760">So</span> <span m="4426310">what</span> <span m="4426470">strategy</span> <span m="4426980">do</span> <span m="4427050">you</span> <span m="4427130">think</span> <span m="4427280">we're</span> <span m="4427380">going</span> <span m="4427480">to</span> <span m="4427530">use</span> <span m="4427780">for</span> <span m="4427860">the</span> <span m="4427950">proof?</span> </p>
<p></p>
<p><span m="4429364">AUDIENCE: Strong induction.</span> </p>
<p><span m="4430270">PROFESSOR: Strong</span> <span m="4430760">induction--</span> <span m="4431490">so</span> <span m="4431790">you</span> <span m="4431860">want</span> <span m="4431950">to</span> <span m="4432020">write</span> <span m="4432240">that</span> <span m="4432450">down--</span> <span m="4432880">proof</span> <span m="4433170">by</span> <span m="4433300">strong</span> <span m="4433670">induction.</span> <span m="4442110">What's</span> <span m="4442330">the</span> <span m="4442430">next</span> <span m="4442670">step</span> <span m="4444770">in a</span> <span m="4444900">proof</span> <span m="4445360">by</span> <span m="4446190">any</span> <span m="4446410">kind</span> <span m="4446620">of</span> <span m="4446690">induction?</span> </p>
<p><span m="4447430">AUDIENCE: Base case.</span> </p>
<p><span m="4448650">PROFESSOR: Base</span> <span m="4449310">case--</span> <span m="4449620">not</span> <span m="4449910">yet.</span> </p>
<p><span m="4451290">AUDIENCE: The predicate.</span> </p>
<p><span m="4452230">PROFESSOR: The predicate--</span> <span m="4453280">what's</span> <span m="4453690">the</span> <span m="4453790">predicate</span> <span m="4454320">here?</span> <span m="4455880">All</span> <span m="4456070">right,</span> <span m="4456360">so</span> <span m="4457890">for</span> <span m="4458130">the</span> <span m="4458230">inductive</span> <span m="4458650">hypothesis,</span> <span m="4459460">the</span> <span m="4459550">predicate--</span> <span m="4461580">any</span> <span m="4461800">guesses</span> <span m="4462260">as</span> <span m="4462390">to</span> <span m="4462490">what</span> <span m="4462690">P</span> <span m="4462870">of</span> <span m="4462970">n is</span> <span m="4463140">going</span> <span m="4463280">to</span> <span m="4463330">be?</span> </p>
<p><span m="4464118">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="4466360">PROFESSOR: What's</span> <span m="4466540">that?</span> </p>
<p><span m="4467288">AUDIENCE: n should be</span> <span m="4467786">2n</span> <span m="4468284">minus 1</span> <span m="4468782">by 2.</span> </p>
<p><span m="4471070">PROFESSOR: Yeah,</span> <span m="4472150">that's</span> <span m="4472590">part</span> <span m="4472960">of</span> <span m="4473080">it,</span> <span m="4473280">but</span> <span m="4474200">we're</span> <span m="4474330">going</span> <span m="4474420">to</span> <span m="4474470">use</span> <span m="4474680">this</span> <span m="4474850">as</span> <span m="4474970">a</span> <span m="4475040">predicate</span> <span m="4475450">to</span> <span m="4475520">start</span> <span m="4475850">with.</span> <span m="4476680">You've</span> <span m="4477020">gone</span> <span m="4477410">a</span> <span m="4477460">step</span> <span m="4477710">ahead</span> <span m="4478060">there.</span> <span m="4479040">So</span> <span m="4479120">I'm</span> <span m="4479240">going</span> <span m="4479330">to</span> <span m="4479400">start</span> <span m="4479730">with</span> <span m="4479980">this</span> <span m="4480210">as</span> <span m="4480300">being</span> <span m="4480520">a</span> <span m="4480580">predicate,</span> <span m="4481090">because</span> <span m="4481260">I'm</span> <span m="4481350">trying</span> <span m="4481570">to</span> <span m="4481650">show</span> <span m="4481850">they're</span> <span m="4482010">all</span> <span m="4482190">the</span> <span m="4482280">same.</span> </p>
<p><span m="4483150">And</span> <span m="4483220">I</span> <span m="4483270">haven't</span> <span m="4483520">figured</span> <span m="4483780">out</span> <span m="4483890">what</span> <span m="4483980">the</span> <span m="4484070">scores</span> <span m="4484420">going</span> <span m="4484530">to</span> <span m="4484580">be.</span> <span m="4484800">But</span> <span m="4484910">I</span> <span m="4484990">think</span> <span m="4485260">you're</span> <span m="4485410">already</span> <span m="4485700">on</span> <span m="4485830">a</span> <span m="4485880">good</span> <span m="4486030">track</span> <span m="4486370">here</span> <span m="4487750">for</span> <span m="4487850">what</span> <span m="4487970">the</span> <span m="4488060">score</span> <span m="4488320">is</span> <span m="4488400">going</span> <span m="4488490">to</span> <span m="4488540">be.</span> <span m="4490410">Now</span> <span m="4490850">we</span> <span m="4490950">do</span> <span m="4491070">the</span> <span m="4491160">base</span> <span m="4491450">case.</span> </p>
<p><span m="4498430">So</span> <span m="4498610">we</span> <span m="4498700">take</span> <span m="4498980">n</span> <span m="4499170">equals</span> <span m="4499470">1--</span> <span m="4500390">the</span> <span m="4500500">one-block</span> <span m="4501280">game.</span> <span m="4502770">S</span> <span m="4503030">of</span> <span m="4503130">1</span> <span m="4503480">is</span> <span m="4503710">0.</span> <span m="4505130">We</span> <span m="4505260">never</span> <span m="4505490">took</span> <span m="4505660">a</span> <span m="4505720">move.</span> <span m="4506170">We</span> <span m="4506300">never</span> <span m="4506520">got</span> <span m="4506680">a</span> <span m="4506740">score.</span> <span m="4507080">There</span> <span m="4507190">was</span> <span m="4507320">only</span> <span m="4507540">one</span> <span m="4507750">block.</span> <span m="4508150">We</span> <span m="4508510">ended</span> <span m="4508730">before</span> <span m="4508990">we</span> <span m="4509100">started.</span> <span m="4510560">So</span> <span m="4510670">that's</span> <span m="4510930">OK.</span> </p>
<p><span m="4513680">And</span> <span m="4513810">now</span> <span m="4514020">we</span> <span m="4514120">have</span> <span m="4514330">the</span> <span m="4514430">inductive</span> <span m="4514870">step.</span> <span m="4518660">And</span> <span m="4518830">here</span> <span m="4519150">we</span> <span m="4519280">get</span> <span m="4519440">to</span> <span m="4519550">assume</span> <span m="4522570">P1,</span> <span m="4525170">P2,</span> <span m="4526700">all</span> <span m="4527020">the</span> <span m="4527110">way</span> <span m="4527270">up</span> <span m="4527410">to</span> <span m="4527500">P n,</span> <span m="4529510">to</span> <span m="4529970">prove</span> <span m="4530520">P n</span> <span m="4530880">plus</span> <span m="4531120">1.</span> <span m="4536020">And</span> <span m="4536300">so</span> <span m="4536400">now</span> <span m="4536660">we</span> <span m="4536810">look</span> <span m="4537050">at</span> <span m="4537350">n</span> <span m="4537580">plus</span> <span m="4537850">1</span> <span m="4538060">blocks,</span> <span m="4541990">because</span> <span m="4542190">that's</span> <span m="4542410">we're</span> <span m="4542570">starting.</span> </p>
<p><span m="4548210">So</span> <span m="4548370">we</span> <span m="4548450">have</span> <span m="4548580">n</span> <span m="4548730">plus</span> <span m="4549020">1,</span> <span m="4549420">and</span> <span m="4549510">we</span> <span m="4549620">split</span> <span m="4549980">it</span> <span m="4550150">somehow.</span> <span m="4551890">And</span> <span m="4552290">it</span> <span m="4552370">could</span> <span m="4552510">be</span> <span m="4552640">any</span> <span m="4552870">split.</span> <span m="4553500">We</span> <span m="4553600">might</span> <span m="4553780">have</span> <span m="4553930">K</span> <span m="4554240">on</span> <span m="4554340">one</span> <span m="4554540">side,</span> <span m="4555480">then</span> <span m="4555670">n</span> <span m="4555920">plus</span> <span m="4556240">1</span> <span m="4556490">minus</span> <span m="4556970">K</span> <span m="4557700">on</span> <span m="4557840">the</span> <span m="4557960">other</span> <span m="4558160">side</span> <span m="4559690">for</span> <span m="4559830">any</span> <span m="4560130">K</span> <span m="4560750">between</span> <span m="4561090">1</span> <span m="4561240">and n.</span> <span m="4565630">And</span> <span m="4565810">let's</span> <span m="4566010">figure</span> <span m="4566320">out</span> <span m="4566680">our</span> <span m="4566910">score</span> <span m="4567990">for</span> <span m="4568090">that</span> <span m="4568370">scenario.</span> </p>
<p><span m="4572600">Well,</span> <span m="4572890">we</span> <span m="4573010">get</span> <span m="4573800">the</span> <span m="4573920">product</span> <span m="4574460">of</span> <span m="4574540">these</span> <span m="4574810">numbers</span> <span m="4575170">for</span> <span m="4575260">the</span> <span m="4575360">first</span> <span m="4575640">step.</span> <span m="4578060">We</span> <span m="4578200">get</span> <span m="4578460">K</span> <span m="4579200">times</span> <span m="4579650">n</span> <span m="4579950">plus</span> <span m="4580290">1</span> <span m="4580550">minus</span> <span m="4580940">K</span> <span m="4581160">for</span> <span m="4581280">the</span> <span m="4581370">first</span> <span m="4581690">step.</span> <span m="4584100">And</span> <span m="4584290">then</span> <span m="4584470">as</span> <span m="4584610">we</span> <span m="4584830">recurse,</span> <span m="4585580">and</span> <span m="4585800">we</span> <span m="4587250">split</span> <span m="4587570">up</span> <span m="4587750">that</span> <span m="4587960">block</span> <span m="4588230">of</span> <span m="4588330">K</span> <span m="4588550">blocks,</span> <span m="4589750">how</span> <span m="4589990">many</span> <span m="4590750">points</span> <span m="4591120">do</span> <span m="4591210">we</span> <span m="4591300">get</span> <span m="4591640">for</span> <span m="4591870">splitting</span> <span m="4592240">that</span> <span m="4592380">all</span> <span m="4592550">the</span> <span m="4592630">way</span> <span m="4592750">down?</span> </p>
<p><span m="4594030">AUDIENCE: P of K.</span> </p>
<p><span m="4594900">PROFESSOR: P of K,</span> <span m="4596340">because</span> <span m="4596730">that's</span> <span m="4596980">the</span> <span m="4597100">induction.</span> <span m="4597610">We</span> <span m="4597750">assume</span> <span m="4598010">that</span> <span m="4598110">as</span> <span m="4598230">part of the</span> <span m="4598420">induction</span> <span m="4598820">hypothesis.</span> <span m="4602700">And</span> <span m="4602910">how</span> <span m="4603240">many</span> <span m="4603680">points</span> <span m="4604140">do</span> <span m="4604230">we</span> <span m="4604310">get</span> <span m="4604580">for</span> <span m="4604710">splitting</span> <span m="4605080">this</span> <span m="4605230">stack</span> <span m="4605610">all</span> <span m="4605790">the</span> <span m="4605860">way</span> <span m="4606000">down?</span> <span m="4608940">Yeah,</span> <span m="4609250">P</span> <span m="4609590">of</span> <span m="4609870">n</span> <span m="4610040">plus</span> <span m="4610350">1</span> <span m="4610826">minus K.</span> </p>
<p><span m="4615590">And</span> <span m="4615800">you</span> <span m="4615950">see</span> <span m="4616210">why</span> <span m="4616390">strong</span> <span m="4616810">induction's</span> <span m="4617330">coming</span> <span m="4617640">in</span> <span m="4617720">handy</span> <span m="4618130">here?</span> <span m="4620310">For</span> <span m="4620390">the</span> <span m="4620500">TA's</span> <span m="4620890">strategy,</span> <span m="4621440">K</span> <span m="4621680">would</span> <span m="4621820">be</span> <span m="4621930">1.</span> <span m="4622990">We</span> <span m="4623120">know</span> <span m="4623310">P</span> <span m="4623530">of</span> <span m="4623600">1</span> <span m="4623820">is</span> <span m="4623970">0.</span> <span m="4625280">And</span> <span m="4625450">you</span> <span m="4625550">would</span> <span m="4625650">just</span> <span m="4625890">need</span> <span m="4626050">ordinary</span> <span m="4626470">induction.</span> <span m="4626890">This would</span> <span m="4626980">be</span> <span m="4627295">P of</span> <span m="4627610">n.</span> </p>
<p><span m="4628760">But</span> <span m="4628910">for</span> <span m="4629030">a</span> <span m="4629080">general</span> <span m="4629470">strategy,</span> <span m="4629915">it</span> <span m="4630360">could</span> <span m="4630500">be</span> <span m="4630610">any</span> <span m="4630820">split.</span> <span m="4631300">So</span> <span m="4631450">we</span> <span m="4631530">need</span> <span m="4631680">to</span> <span m="4631740">have</span> <span m="4631910">all</span> <span m="4632170">these</span> <span m="4632370">assumed</span> <span m="4632740">here.</span> <span m="4633110">It</span> <span m="4633340">makes</span> <span m="4633550">it</span> <span m="4633630">much</span> <span m="4633880">easier</span> <span m="4634870">when</span> <span m="4634960">using</span> <span m="4635280">strong</span> <span m="4635590">induction.</span> </p>
<p><span m="4638330">All</span> <span m="4638500">right,</span> <span m="4638810">so</span> <span m="4638900">this</span> <span m="4639100">is</span> <span m="4639220">our</span> <span m="4639360">total</span> <span m="4639730">score</span> <span m="4640310">for</span> <span m="4640450">the</span> <span m="4640600">game</span> <span m="4641270">for</span> <span m="4641480">n</span> <span m="4641560">plus</span> <span m="4641770">1</span> <span m="4641940">blocks.</span> <span m="4643180">And</span> <span m="4643340">we're</span> <span m="4643450">trying</span> <span m="4643840">to</span> <span m="4643900">show</span> <span m="4644220">it</span> <span m="4646170">just</span> <span m="4646340">depends</span> <span m="4646660">on</span> <span m="4646790">n,</span> <span m="4648240">and</span> <span m="4648390">that</span> <span m="4648490">it</span> <span m="4648580">doesn't</span> <span m="4648870">depend</span> <span m="4649200">on</span> <span m="4649330">K.</span> <span m="4651201">Does</span> <span m="4651688">that</span> <span m="4652175">depend</span> <span m="4652662">on K?</span> </p>
<p><span m="4658030">Looks</span> <span m="4658340">like</span> <span m="4658480">it</span> <span m="4658550">to</span> <span m="4658640">me,</span> <span m="4658890">right?</span> <span m="4660210">I</span> <span m="4660300">got</span> <span m="4660530">K</span> <span m="4660770">in</span> <span m="4660880">every</span> <span m="4661080">single</span> <span m="4661400">term.</span> <span m="4664020">So</span> <span m="4664050">I'm</span> <span m="4664170">not</span> <span m="4664360">there.</span> <span m="4664740">I'm</span> <span m="4664830">stuck.</span> </p>
<p><span m="4667720">You</span> <span m="4667790">see</span> <span m="4667920">why</span> <span m="4668060">I'm</span> <span m="4668160">stuck?</span> <span m="4668940">Because I'd</span> <span m="4669050">like to</span> <span m="4669280">be able</span> <span m="4669470">to</span> <span m="4669530">say</span> <span m="4670380">that</span> <span m="4670510">the</span> <span m="4670590">score</span> <span m="4670970">I</span> <span m="4671030">get</span> <span m="4671230">for</span> <span m="4671310">the</span> <span m="4671400">n</span> <span m="4671520">plus</span> <span m="4671720">one</span> <span m="4671950">block</span> <span m="4672240">game--</span> <span m="4673950">this</span> <span m="4674120">is</span> <span m="4674240">S n</span> <span m="4674580">plus</span> <span m="4674810">1--</span> <span m="4676210">is</span> <span m="4676420">the</span> <span m="4676530">same</span> <span m="4677060">for</span> <span m="4677230">any</span> <span m="4677640">sequence</span> <span m="4678020">of</span> <span m="4678100">moves.</span> <span m="4678520">It doesn't</span> <span m="4678880">even depend</span> <span m="4679200">on</span> <span m="4679410">K.</span> <span m="4680500">But</span> <span m="4680620">I</span> <span m="4680790">can't</span> <span m="4681060">do</span> <span m="4681170">that.</span> </p>
<p><span m="4682260">What</span> <span m="4682420">do</span> <span m="4682460">you</span> <span m="4682580">do</span> <span m="4682830">when</span> <span m="4682950">you're</span> <span m="4683080">stuck</span> <span m="4683420">with an</span> <span m="4683540">induction</span> <span m="4683980">proof?</span> <span m="4685310">What's</span> <span m="4685510">one</span> <span m="4685650">of</span> <span m="4685690">the</span> <span m="4685780">tricks?</span> </p>
<p><span m="4686460">AUDIENCE: Make it</span> <span m="4686520">stronger.</span> </p>
<p><span m="4687370">PROFESSOR: Make</span> <span m="4687660">it</span> <span m="4687790">stronger,</span> <span m="4689610">which</span> <span m="4689800">means</span> <span m="4690060">I</span> <span m="4690180">get</span> <span m="4690390">a</span> <span m="4690470">stronger</span> <span m="4691240">induction</span> <span m="4691590">hypothesis.</span> <span m="4693360">Now,</span> <span m="4693390">somebody</span> <span m="4693850">gave</span> <span m="4694180">me</span> <span m="4694320">one</span> <span m="4694650">up</span> <span m="4694870">there</span> <span m="4696050">before.</span> <span m="4697360">How</span> <span m="4697620">could</span> <span m="4697750">I</span> <span m="4697820">strengthen</span> <span m="4698330">my</span> <span m="4698510">induction</span> <span m="4698660">hypothesis</span> <span m="4698720">here?</span> </p>
<p><span m="4702052">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="4703960">PROFESSOR: Give</span> <span m="4704230">the</span> <span m="4704320">formula</span> <span m="4705650">for</span> <span m="4706010">S n--</span> <span m="4707120">that</span> <span m="4707320">would</span> <span m="4707430">make</span> <span m="4707600">it</span> <span m="4707670">stronger.</span> <span m="4708360">I</span> <span m="4708500">tell</span> <span m="4708820">you</span> <span m="4708960">what</span> <span m="4709100">it</span> <span m="4709190">is.</span> <span m="4710140">What's</span> <span m="4710360">a</span> <span m="4710420">good</span> <span m="4710630">guess</span> <span m="4710860">for</span> <span m="4710930">the</span> <span m="4711010">formula</span> <span m="4711470">here?</span> </p>
<p><span m="4712760">AUDIENCE: Factorial.</span> </p>
<p><span m="4713700">PROFESSOR: Factorial--</span> <span m="4715780">Well,</span> <span m="4717290">is</span> <span m="4717970">S of 8</span> <span m="4718415">equal</span> <span m="4718860">8</span> <span m="4719110">factorial?</span> <span m="4721580">No,</span> <span m="4722630">not</span> <span m="4722930">so</span> <span m="4722990">good.</span> </p>
<p><span m="4724234">[SIDE CONVERSATION]</span> </p>
<p><span m="4726669">AUDIENCE: n equals</span> <span m="4727156">n  minus 1</span> <span m="4727643">over 2.</span> </p>
<p><span m="4728130">PROFESSOR: Yeah,</span> <span m="4728580">that's</span> <span m="4728800">a better</span> <span m="4729120">guess--</span> <span m="4731510">equals</span> <span m="4732200">n</span> <span m="4733310">times</span> <span m="4733600">n</span> <span m="4733750">minus</span> <span m="4734110">1</span> <span m="4734380">over</span> <span m="4734680">2.</span> <span m="4736160">And</span> <span m="4736280">in</span> <span m="4736360">fact,</span> <span m="4737800">what's</span> <span m="4738240">8</span> <span m="4738710">times</span> <span m="4739010">7</span> <span m="4739360">over</span> <span m="4739540">2?</span> <span m="4739750">It's</span> <span m="4739850">28.</span> <span m="4740330">It</span> <span m="4740390">works</span> <span m="4740720">here.</span> </p>
<p><span m="4741910">Let's</span> <span m="4742090">check</span> <span m="4742300">another</span> <span m="4742610">one.</span> <span m="4742770">What's</span> <span m="4743740">S of</span> <span m="4744110">2?</span> </p>
<p><span m="4747220">AUDIENCE: 1.</span> </p>
<p><span m="4747670">PROFESSOR: 1--</span> <span m="4749430">well,</span> <span m="4749650">2</span> <span m="4749950">times</span> <span m="4750220">1</span> <span m="4750390">over</span> <span m="4750540">2--</span> <span m="4750825">it</span> <span m="4751110">works.</span> <span m="4752740">What's</span> <span m="4753140">S</span> <span m="4753566">of</span> <span m="4754420">3?</span> <span m="4757600">2</span> <span m="4757960">the</span> <span m="4758050">first</span> <span m="4758420">move,</span> <span m="4759530">then</span> <span m="4759690">1</span> <span m="4759930">more.</span> <span m="4761130">3--</span> <span m="4762116">3</span> <span m="4762610">times</span> <span m="4762870">2</span> <span m="4763010">over</span> <span m="4763180">2</span> <span m="4763390">is</span> <span m="4763570">3--</span> <span m="4764590">looks</span> <span m="4764820">good.</span> <span m="4766340">So</span> <span m="4766500">now</span> <span m="4766630">I've</span> <span m="4766770">got</span> <span m="4766890">a</span> <span m="4766930">stronger</span> <span m="4767420">induction</span> <span m="4767740">hypothesis.</span> <span m="4768430">Not</span> <span m="4768670">only</span> <span m="4768890">is it</span> <span m="4769040">always</span> <span m="4769350">the</span> <span m="4769460">same,</span> <span m="4770080">it is</span> <span m="4770640">that</span> <span m="4770930">number</span> <span m="4771500">is</span> <span m="4771610">your</span> <span m="4771750">score.</span> </p>
<p><span m="4773710">Let's</span> <span m="4773950">see</span> <span m="4774050">if</span> <span m="4774140">that</span> <span m="4774330">works.</span> <span m="4775190">Let's</span> <span m="4775370">plug</span> <span m="4775620">that</span> <span m="4775800">in</span> <span m="4775940">now.</span> <span m="4778000">So</span> <span m="4778580">base</span> <span m="4778860">case--</span> <span m="4779200">let's</span> <span m="4779390">check</span> <span m="4779670">it</span> <span m="4779740">out--</span> <span m="4780560">is</span> <span m="4781610">1</span> <span m="4782170">times</span> <span m="4782490">1</span> <span m="4782740">minus</span> <span m="4783130">1</span> <span m="4783530">over</span> <span m="4783880">2</span> <span m="4785120">equal</span> <span m="4785430">to</span> <span m="4785500">0.</span> <span m="4787520">Yes.</span> </p>
<p><span m="4789750">Let's</span> <span m="4789980">plug</span> <span m="4790220">it</span> <span m="4790280">in</span> <span m="4790390">here</span> <span m="4790630">now.</span> <span m="4793040">P of</span> <span m="4793480">K</span> <span m="4793920">would</span> <span m="4794060">be</span> <span m="4794190">now</span> <span m="4794600">K,</span> <span m="4795526">K</span> <span m="4795990">minus</span> <span m="4796390">1</span> <span m="4796620">over</span> <span m="4796830">2.</span> <span m="4798570">This</span> <span m="4798830">would</span> <span m="4798980">be</span> <span m="4799300">n</span> <span m="4800070">plus</span> <span m="4800310">1</span> <span m="4800580">minus</span> <span m="4800950">K,</span> <span m="4802702">n</span> <span m="4803140">minus</span> <span m="4803520">K</span> <span m="4804560">over</span> <span m="4804880">2</span> <span m="4807520">plus</span> <span m="4808050">this</span> <span m="4808260">one</span> <span m="4808430">here.</span> </p>
<p><span m="4812140">All</span> <span m="4812260">right,</span> <span m="4813060">I've</span> <span m="4813170">got</span> <span m="4813330">to</span> <span m="4813650">add</span> <span m="4813950">all</span> <span m="4814210">these</span> <span m="4814430">things</span> <span m="4814720">up,</span> <span m="4815490">and</span> <span m="4815640">see</span> <span m="4815850">if</span> <span m="4815950">it</span> <span m="4816010">equals</span> <span m="4816630">that</span> <span m="4816820">expression</span> <span m="4817340">for</span> <span m="4817480">n</span> <span m="4817570">plus</span> <span m="4817800">1.</span> <span m="4819473">Let's</span> <span m="4819954">do that.</span> <span m="4823330">All</span> <span m="4823510">right,</span> <span m="4823710">I</span> <span m="4823780">get</span> <span m="4824190">2</span> <span m="4824420">K n</span> <span m="4826510">plus</span> <span m="4826980">2</span> <span m="4827200">K</span> <span m="4828370">minus</span> <span m="4828720">2</span> <span m="4828960">K</span> <span m="4829220">squared.</span> <span m="4829780">I'm</span> <span m="4829860">putting</span> <span m="4830060">all</span> <span m="4830210">this</span> <span m="4830370">over</span> <span m="4830530">2--</span> <span m="4831750">one</span> <span m="4832040">giant</span> <span m="4832580">mess</span> <span m="4832900">over</span> <span m="4833100">2</span> <span m="4833320">here--</span> <span m="4834920">plus</span> <span m="4835470">K</span> <span m="4835700">squared</span> <span m="4836790">minus</span> <span m="4837250">K</span> <span m="4840220">plus</span> <span m="4840560">n</span> <span m="4840750">squared.</span> <span m="4843120">Well,</span> <span m="4843330">let's</span> <span m="4843520">write</span> <span m="4843800">it</span> <span m="4843910">just</span> <span m="4844190">as</span> <span m="4845560">n</span> <span m="4845810">plus</span> <span m="4846110">1</span> <span m="4846300">times</span> <span m="4846736">n,</span> <span m="4851030">minus</span> <span m="4851415">K</span> <span m="4851800">n,</span> <span m="4852940">minus</span> <span m="4853400">K,</span> <span m="4854950">minus</span> <span m="4855380">K</span> <span m="4855675">n,</span> <span m="4856520">all</span> <span m="4856820">over</span> <span m="4857010">2.</span> </p>
<p><span m="4857860">And</span> <span m="4857970">now</span> <span m="4858140">let's</span> <span m="4858360">cancel.</span> <span m="4859770">Two</span> <span m="4859960">K</span> <span m="4860240">n's</span> <span m="4860480">here</span> <span m="4861060">cancel</span> <span m="4862050">K n,</span> <span m="4862480">K</span> <span m="4862830">n.</span> <span m="4864420">2K</span> <span m="4865180">cancels</span> <span m="4865760">K</span> <span m="4866880">and</span> <span m="4867070">K.</span> <span m="4869776">2K</span> <span m="4870560">squared</span> <span m="4871620">minus</span> <span m="4871830">2</span> <span m="4871940">K</span> <span m="4872070">squared</span> <span m="4872310">cancels</span> <span m="4872710">K</span> <span m="4872910">squared.</span> <span m="4874070">And</span> <span m="4874460">I</span> <span m="4874660">forgot,</span> <span m="4875560">plus</span> <span m="4877560">K</span> <span m="4877720">squared</span> <span m="4878790">from</span> <span m="4878960">here</span> <span m="4879170">and</span> <span m="4879230">here.</span> </p>
<p><span m="4880970">So</span> <span m="4881090">I'm</span> <span m="4881240">left</span> <span m="4881610">with</span> <span m="4882300">n</span> <span m="4882500">plus</span> <span m="4882760">1</span> <span m="4882960">times</span> <span m="4883230">n</span> <span m="4883420">over</span> <span m="4883640">2.</span> <span m="4885040">That</span> <span m="4885330">is</span> <span m="4885650">S n</span> <span m="4886020">plus</span> <span m="4886240">1.</span> <span m="4889080">And</span> <span m="4889140">the</span> <span m="4889230">K</span> <span m="4889710">disappears.</span> </p>
<p><span m="4890770">We've</span> <span m="4890930">established,</span> <span m="4891630">now,</span> <span m="4891820">the</span> <span m="4891920">stronger</span> <span m="4893040">induction</span> <span m="4893460">hypothesis.</span> <span m="4894740">So</span> <span m="4894890">not</span> <span m="4895160">only</span> <span m="4895360">did</span> <span m="4895460">we</span> <span m="4895580">prove</span> <span m="4895980">that</span> <span m="4896140">every</span> <span m="4897090">set</span> <span m="4897280">of</span> <span m="4898330">unstackings</span> <span m="4898980">gives</span> <span m="4899120">you</span> <span m="4899200">the</span> <span m="4899300">same</span> <span m="4899580">score.</span> <span m="4900960">We</span> <span m="4901110">proved</span> <span m="4901580">the</span> <span m="4901670">score</span> <span m="4902840">is,</span> <span m="4903460">for n</span> <span m="4903690">blocks,</span> <span m="4904080">n</span> <span m="4904430">times</span> <span m="4904680">n</span> <span m="4904810">minus</span> <span m="4905120">1</span> <span m="4906010">over</span> <span m="4906250">2.</span> </p>
<p><span m="4910120">That's</span> <span m="4910300">was</span> <span m="4910400">pretty</span> <span m="4910610">good</span> <span m="4910820">for</span> <span m="4910930">strong</span> <span m="4911210">induction.</span> <span m="4911620">It lets you</span> <span m="4911840">do</span> <span m="4911950">some</span> <span m="4912080">pretty</span> <span m="4912320">powerful</span> <span m="4912720">things.</span> <span m="4913710">OK,</span> <span m="4914010">very</span> <span m="4914270">good,</span> <span m="4914460">that's</span> <span m="4914950">it</span> <span m="4915150">for</span> <span m="4915270">today.</span> </p>
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